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Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 47
Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/platne/serwer318421/public_html/zeszyt.jedlikowski.com/wp-content/plugins/latex/latex.php on line 49
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Algebra boolowska została stworzona przez George'a Boole'a w pracach Matematyczna analiza logiki (1847) oraz Rozważania nad prawami myśli (1854). Stała się podstawą rozwoju elektroniki cyfrowej i jest obecna we wszystkich współczesnych językach programowania. Jest także używana w matematycznej teorii zbiorów czy statystyce.
Wartości
Zwykle w algebrze definiujemy liczby (np. 0, 1, 7, …), jednak w algebrze Boole'a definiuje się jedynie wartości prawdy: prawdę oraz fałsz. Wartości te można reprezentować za pomocą bitów (lub: liczb binarnych), odpowiednio 1 oraz 0. Wartości te nie zachowują się jak liczby całkowite, dlatego należy poznać sposób pracy, liczenia, w tej algebrze.
Podstawowe operacje
Podstawowymi operacjami logiki boolowskiej są: koniunkcja (AND, iloczyn logiczny), alternatywa (OR, suma logiczna) oraz negacja (NOT). Operatory te omówiono poniżej.
Koniunkcja
Koniunkcję reprezentuje się za pomocą znaków lub
. Jej działanie reprezentuje niniejsza tabela prawdy.
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Zauważmy sposób działania iloczynu logicznego: , gdy
, w przeciwnym wypadku —
.
Alternatywa
Alternatywę reprezentuje się za pomocą znaków lub
. Jej działanie reprezentuje niniejsza tabela prawdy.
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Zauważmy sposób działania sumy logicznego: , gdy
, w przeciwnym wypadku —
, czyli prawda, gdy którakolwiek przesłanka jest prawdziwa.
Negacja
Negację reprezentuje się za pomocą znaków lub linią ponad znakiem (
). Jej działanie reprezentuje niniejsza tabela prawdy.
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Zauważamy sposób działania negacji: zmiana wartości z prawdy na fałsz i odwrotnie.
Prawa algebry logicznej
Jak w każdej algebrze, i w tej określono pewne zasady. Są nimi łączność, rozdzielność, przemienność oraz prawa De Morgana. Pierwsze trzy dotyczą wyłącznie operatorów AND i OR.
Łączność
Łączność jest taką własnością algebry, że kolejność wykonywania działań nie ma znaczenia (uwaga! tylko przy takim samym operatorze).
Rozdzielność
Własność rozdzielności pozwala zastosować operator na wartości (operandy) w nawiasach.
(rozdzielność koniunkcji względem alternatywy
lub
rozdzielność mnożenia względem dodawania)
(rozdzielność alternatywy względem koniunkcji
lub
rozdzielność dodawania względem mnożenia, przy czym nie można takiej reguły zastosować w „normalnej” matematyce)
Przemienność
Przemienność oznacza, że kolejność stosowania operatora nie ma znaczenia.
Prawa De Morgana
Dziewiętnastowieczny matematyk, Augustus De Morgan, zaproponował używane do dziś następujące prawa definiujące koniunkcję i alternatywę przy pomocy tych funkcji na wzajem oraz negacji. Przedstawia się je jako następujące „regułki”:
Negacja koniunkcji jest alternatywą negacji.
Negacja alternatywy jest koniunkcją negacji.
Oczywiście możemy zaprezentować także przy ich pomocy same funkcje, a nie ich negacje. Negując obie strony równań, otrzymamy:
Ważność operatorów
Należy wskazać:
można ułatwić zrozumienie powyższego zapisu, używając następującego:
Widzimy teraz analogię do znanej z „codziennej” matematyki kolejności wykonywania działań.
Zasady
Dzięki wprowadzonym wcześniej prawom, możemy przyspieszyć pracę z logiką, stosując następujące własności algebry Boole'a.
,
,
– identyczność koniunkcji — wynik będzie zależeć tylko od A, ponieważ 1 nie wpływa na wynik iloczynu logicznego,
– anihilator alternatywy — coś lub prawda to prawda,
– anihilator koniunkcji — coś oraz fałsz to fałsz,
– identyczność alternatywy — coś lub 0 to coś, wynik zależy tylko od A,
– skoro koniunkcja daje prawdę tylko dla
, to nie jest możliwe uzyskanie wyniku innego niż 0 – jeśli A to 1, to negacja 1 da 0, więc otrzymamy
,
– skoro alternatywa daje prawdę, gdy co najmniej jeden operand (term) jest prawdziwy, to zawsze otrzymamy
lub
,
– podwójna negacja „znosi się”,
,
,
Źródła
Boolean Algebra. Wikibooks, open books for an open world. https://en.wikibooks.org/wiki/Electronics/Boolean_Algebra
Boolean Algebra. Wikipedia. https://en.wikipedia.org/wiki/Boolean_algebra
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