{"id":758,"date":"2012-12-17T05:00:38","date_gmt":"2012-12-17T13:00:38","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=758"},"modified":"2012-12-17T05:02:22","modified_gmt":"2012-12-17T13:02:22","slug":"iff-as-conditional-chain","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/12\/iff-as-conditional-chain\/","title":{"rendered":"Iff as Conditional Chain"},"content":{"rendered":"<p>I'm not sure I like how when we want to prove that <em>two<\/em> statements are equivalent, we typically say &quot;<em>A<\/em> if and only if <em>B<\/em>&quot; and we prove it by separately proving &quot;both directions&quot; <em>A<\/em> \u21d2 <em>B<\/em> and <em>B<\/em> \u21d2 <em>A<\/em>, but when we want to prove three or more statements are equivalent, we typically say &quot;The following are equivalent&quot; and prove a &quot;circular chain&quot; of conditionals (1) \u21d2 (2) \u21d2 [...] \u21d2 (n) \u21d2 (1), as if these were <em>different<\/em> proof strategies. Because really, the &quot;both directions&quot; business is just a special case of the chain-of-conditionals idea: (1) \u21d2 (2) \u21d2 (1). At the very least, one of my books ought to have mentioned this.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I'm not sure I like how when we want to prove that two statements are equivalent, we typically say &quot;A if and only if B&quot; and we prove it by separately proving &quot;both directions&quot; A \u21d2 B and B \u21d2 &hellip; <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/12\/iff-as-conditional-chain\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[45],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/758"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=758"}],"version-history":[{"count":4,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/758\/revisions"}],"predecessor-version":[{"id":763,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/758\/revisions\/763"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=758"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=758"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}