[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-398591-105":59,"doc-detail-398591-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","davids-sorta-super-quick-logic-cheat-sheet","David’s (Sorta) Super Quick Logic Cheat Sheet","","A concise learning guide to propositional logic for proving mathematical statements. It defines propositions, explains how propositions can involve variables, and presents the main formation rules for combining them: conjunction (and), disjunction (or), implication (if-then), universal quantification (for all), existential quantification (there exists), and negation (not), including biconditional meaning. It also introduces basic propositions used as building blocks, such as membership, function definitions, and equality, then transitions to the first step for proving using assumptions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/davids-sorta-super-quick-logic-cheat-sheet/398591/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/davids-sorta-super-quick-logic-cheat-sheet/398591.png","ImageObject",300,407,{"name":92,"@type":93},"kopisore","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-21","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is a proposition in this logic cheat sheet?","Question",{"text":112,"@type":113},"A proposition is a statement that you could argue is true or false. It may involve variable elements, such as an equation in x or a predicate about sets.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Which formation rules are used to build new propositions?",{"text":117,"@type":113},"The sheet lists conjunction (A ∧ B), disjunction (A ∨ B), implication (A ⇒ B), universal quantification (∀x: A(x)), existential quantification (∃x: A(x)), negation (not A), and biconditional meaning (“A if and only if B”).",{"name":119,"@type":110,"acceptedAnswer":120},"What are the basic building blocks of propositions mentioned in the document?",{"text":121,"@type":113},"It highlights fundamental propositions such as membership (a ∈ A), function mapping notation (f: A → B and f(a)=b), and equality (x = y) with meaning depending on whether the objects are numbers, sets, or functions.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},398591,1792111577,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":29,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144,"preview_page_count":4},962090880963,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","David’s (Sorta) Super Quick Logic Cheat Sheet .  \nOk, so you need to prove something. Take a deep breath. What do you need to prove? 1  \nPropositions  \nA proposition is something you are saying that you could, potentially, argue is true or correct (or, alternatively, argue is false or incorrect) .  \nEvery proposition is mathematics is built out of smaller, less scary propositions. Propositions might be something like “6 is even”, or “if x ∈ R, then 2x > 0”, or “A subset C ⊆ X is closed if and only if every convergent sequence in C converges to a point in C”. Don’t worry if you don’t know what these phrases mean – we’ll learn about them in this course!  \nLuckily for us, there is only a small list of ways to combine propositions that are enough to say everything in mathematics! In the following, I will use fancy letters A, B , C etc to denote propositions. For example, A might be the proposition “2 + 2 = 4”, and B might be the proposition “3 divides 13”. If the proposition isabout a variable element, like the Proposition “x + 4 = 14”, I will write A (x) to signal that the proposition A is about a variable x. Ready? Here are all the combinations:  \n(And) If A and B are propositions, then “A and B” is a proposition. We write in symbolic shorthand as A ∧ B.  \n(Or) If A and B are propositions, then “A or B” is a proposition. We write this in symbolic shorthand as A ∨ B.  \n(If-Then) If A and B are propositions, then “if A, then B” is a proposition. We also sometimes say “A implies B” or even “B if A”. We write this in symbolic shorthand as A ⇒ B.  \n(For-All) If A(x) is a proposition about a variable x, then “for all x, A (x)” is a proposition. We also sometimes say “for any” or “for each”.2 We write this in symbolic shorthand as ∀x: A (x) .  \n(Exists) If A(x) is a proposition about a variable x, then “there exists an x such that A (x)” is a proposition. We also say “there is an x such that A(x)”, or “for some x, A (x)”. We write this in symbolic shorthand as ∃x: A (x) .3  \n(Not) If A is a proposition, then “not A” is a proposition.  \nFinally, the dreaded “A if and only if B” just means  \n“(if A then B) and (if B then A)”.  \nYou don’t need to know this, but these are generally called the formation rules of the propositions, because they tell you how to make new propositions (“form” new propositions) out of old ones.  \nAside: Basic Propositions  \nThe building blocks of propositions are really basic propositions, usually like x ∈ X (read “x is in X” or “x is a X” depending on context4 )or x = y (“x equals y” or sometimes “x is y”) . There are effectively three basic propositions that are enough to do all of math (at least in principle, if you have lots of time on your hands):  \n1 I’ll be referring to “An Infinite Descent into Pure Mathematics” as “the Book” here. You can find a copy at [https://infinitedescent.xyz/](https://infinitedescent.xyz/. It)[. It](https://infinitedescent.xyz/. It)’s really great, I highly recommend reading along. This material is basically a condensed version of chapter 1 .  \n2 Note that “for all x, A (x)” is no longer about the variable x. This is because it is saying something about all things; the sentence “for all y , A (y)” means the exact same thing (as long as y wasn’t a variable appearing in A to begin with) . In the statement “for all x, A (x)”, we say that x is bound by the quantifier “for all”. Don’t worry if this is confusing, it will make more sense as you work with quantifiers more.  \n3We write “there exists” with a backwards E – 9 – for “exists”, and we write “for all” with an upside down A – 8 – for “all”.  \n4 See Appendix A of the book for a more detailed discussion of how to read these symbols aloud.  \n• If A is a set (that is, a collection of distinct, individual things), then we can say a ∈ A which means that a is an element of the set A (that is, a is in the collection A) . For example, 3 ∈ N is a true propositon: 3 is in the collection N of all natural (or counting) numbers. We’ll do more w","cbCaiaUzyvArnb76","https://ap.wps.com/l/cbCaiaUzyvArnb76","pdf",82098,"English","# Propositions\n## Formation rules (connectives and quantifiers)\n## Aside: Basic propositions\n# Proving things, step 1","[{\"question\":\"What is a proposition in this logic cheat sheet?\",\"answer\":\"A proposition is a statement that you could argue is true or false. It may involve variable elements, such as an equation in x or a predicate about sets.\"},{\"question\":\"Which formation rules are used to build new propositions?\",\"answer\":\"The sheet lists conjunction (A ∧ B), disjunction (A ∨ B), implication (A ⇒ B), universal quantification (∀x: A(x)), existential quantification (∃x: A(x)), negation (not A), and biconditional meaning (“A if and only if B”).\"},{\"question\":\"What are the basic building blocks of propositions mentioned in the document?\",\"answer\":\"It highlights fundamental propositions such as membership (a ∈ A), function mapping notation (f: A → B and f(a)=b), and equality (x = y) with meaning depending on whether the objects are numbers, sets, or functions.\"}]","David’s (Sorta) Super Quick Logic Cheat Sheet | PDF",1790485454,15]