[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572267-105":53,"doc-detail-572267-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","math-135-cheat-sheet-for-final-exam","Math 135 Cheat Sheet for Final Exam","","Compact cheat sheet for the Math 135 final exam, organized by key discrete math topics and core results. Covers set theory notation and operations (subset, union, intersection, complement, Cartesian product, power set, cardinality), logical notation and quantifiers, function definitions including injective/surjective/bijective properties and inverses, recurrence-tree intuition and the Master Theorem cases, asymptotic notation with algebraic properties, logarithm and exponent identities, counting formulas such as permutations/combinations and the binomial theorem, and fundamental graph theory terms and formulas including Eulerian concepts, clique/independent sets, bipartite graphs, and Euler’s formula for planar graphs.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/math-135-cheat-sheet-for-final-exam/572267/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/math-135-cheat-sheet-for-final-exam/572267.png","ImageObject",442,249,{"name":88,"@type":89},"EvelynWrites","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What set theory symbols and operations are summarized in the cheat sheet?","Question",{"text":109,"@type":110},"It lists common notation such as empty set, subset/proper subset, superset/proper superset, set equality, union, intersection, difference, symmetric difference, complement, Cartesian product, and power set, along with cardinality.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How are quantifiers and implications written in the logic section?",{"text":114,"@type":110},"It defines proposition and predicate logic, gives the notation for universal quantifier (for all, written 8) and existential quantifier (there exists, written 9), and shows implication and related forms like inverse, converse, and contrapositive.",{"name":116,"@type":107,"acceptedAnswer":117},"Which key tools for recurrence relations and asymptotic growth are provided?",{"text":118,"@type":110},"It includes a recursion tree idea for T(n)=aT(n/b)+f(n) and the Master Theorem cases for comparing a and b^k, plus standard asymptotic notation (o, O, Θ, Ω, ω/! in the sheet) and their inequality-based definitions and combination rules.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572267,1792015510,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":76,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":9},687212989558,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Math 135 Cheat Sheet for Final Exam  \n\n| Set Theory Notation |  |  |  |\n| --- | --- | --- | --- |\n| empty set\u003Cbr>subset proper subset\u003Cbr>superset proper superset set equality\u003Cbr>union intersection di􀀋erence symmetric di􀀋erence complement Cartesian product power set\u003Cbr>cardinality | ?\u003Cbr>A 􀀒 B\u003Cbr>A 􀀚 B\u003Cbr>A 􀀓 B\u003Cbr>A 􀀛 B\u003Cbr>A = B\u003Cbr>A [ B\u003Cbr>A \\ B\u003Cbr>A 􀀀 B\u003Cbr>A􀀁  BA\u003Cbr>A 􀀂 BP (A)\u003Cbr>jA j | f g\u003Cbr>8x : x 2 A ! x 2 BA 􀀒 B ^ 9y 2 B : y 26 AB 􀀒 A\u003Cbr>B 􀀚 A |  |\n|  |  | A 􀀒 B ^ B 􀀒 Afx j x 2 A _ x 2 Bg\u003Cbr>fx j x 2 A ^ x 2 Bg |  |\n|  |  | fx j x 2 A ^ x 26 Bg = A \\ B\u003Cbr>fx j x 2 A $ x 26 Bg fx j x 26 Ag = U 􀀀 Af (a; b) j a 2 A ^ b 2 Bg fB j B 􀀒 Ag\u003Cbr>\\# of elements (if 􀀌nite) |  |\n\n\n| Logic |  |\n| --- | --- |\n| proposition\u003Cbr>predicate logical operations universal quanti􀀌er existential quanti􀀌er\u003Cbr>implication inverse of p ! q converse of p ! qcontrapositive of p ! q | statement which is unambiguously true or false proposition which incorporates a variable and ^, or _, not :\u003Cbr>for all, written 8\u003Cbr>there exists, written 9\u003Cbr>if p then q, written p ! q\u003Cbr>:p ! :qq ! p:q ! :p |\n\n\n| Function f: A ! B\u003Cbr>A function f from A to B associates each element a 2 A to exactly one element b 2 B. |  |\n| --- | --- |\n| Notation\u003Cbr>one-to-one (or injective) onto (or surjective) bijection\u003Cbr>inverse f 􀀀1 : B ! A | b = f (a) if b is associated to a\u003Cbr>8a1 ; a2 2 A, if a 1  a2 then f (a1 )  f (a2 )\u003Cbr>8b 2 B , 9a 2 A such that f (a) = b\u003Cbr>one-to-one and onto\u003Cbr>f (b; a) j b = f (a)g (if f is a bijection) |\n\n\n| Recursion tree for T (n) = aT (n=b) + f(n) |  |\n| --- | --- |\n| f (n)\u003Cbr>\u003Cbr>f (n=b) : : : f (n=b)\u003Cbr>\u003Cbr>f (n=b2 )\u003Cbr>.\u003Cbr>.\u003Cbr>.\u003Cbr>: : : f (n=b2 )\u003Cbr>.\u003Cbr>.\u003Cbr>.\u003Cbr>f (n=b2 )\u003Cbr>.\u003Cbr>.\u003Cbr>.\u003Cbr>: : : f (n=b2 )\u003Cbr>.\u003Cbr>.\u003Cbr>. | 􀀀! f (n)\u003Cbr>􀀀! a 􀀁 f(n=b)\u003Cbr>􀀀! a2 􀀁 f(n=b2 )\u003Cbr>(logb n levels) |\n| Master Theorem\u003Cbr>Let T(n) = aT (n=b) + O (nk) If a 􀀕 1, b is an integer 􀀕 1, and k a real number 􀀕 0:\u003Cbr>a \u003C bk =) T (n) = O (nk)\u003Cbr>a = bk =) T (n) = O (nk log n)\u003Cbr>a > bk =) T (n) = O (nlog ba) |  |\n\n\n| Asymptotic notation |  |  |\n| --- | --- | --- |\n| f (n) = o(g(n))\u003Cbr>f (n) = O (g(n))\u003Cbr>f (n) = 􀀂(g(n))\u003Cbr>f (n) = 􀀊(g(n))\u003Cbr>f (n) = !(g(n)) | 8c > 0: 9N > 0: 8n 􀀕 N : f (n) \u003C c 􀀁 g(n)\u003Cbr>9c > 0: 9N > 0: 8n 􀀕 N : f (n) 􀀔 c 􀀁 g(n)\u003Cbr>f (n) = O (g(n)) and f (n) = 􀀊(g(n))\u003Cbr>9c > 0: 9N > 0: 8n 􀀕 N : f (n) 􀀕 c 􀀁 g(n)\u003Cbr>8c > 0: 9N > 0: 8n 􀀕 N : f (n) > c 􀀁 g(n) |  |\n| f (n) = O (g(n)) =) f (n) + h(n) = O (g(n) + h(n))\u003Cbr>f (n) = O (g(n)) =) f (n) 􀀁 h(n) = O (g(n) 􀀁 h(n))\u003Cbr>f (n) + g(n) = O (maxff(n); g (n)g) f (n) = O (g(n)) and g(n) = O (h(n)) =) f (n) = O (h(n)) |  |  |\n|  |  |  |\n|  |  | Logarithm identities |\n| 1\u003Cbr>P 􀀋 = 1􀀋~~ ~~ (if 􀀋 \u003C 1) i=0\u003Cbr>d\u003Cbr>P ic = 􀀂(nc+1) (if c  􀀀1) i=0\u003Cbr>n\u003Cbr>P ci = 􀀂(cn) (if c > 1) i=0\u003Cbr>n\u003Cbr>P log i = 􀀂(nlog n) i=1 |  |  |\n|  |  | logb (bx) = x blog b x = x\u003Cbr>logb x = l~~l~~og~~o~~g~~c~~c~~ ~~xb logb (xy) = logb x + logb y logb (1=x) = 􀀀 logb x\u003Cbr>xlog b y = ylogb x logb (xy) = y logb x |\n|  |  |  |\n\n\n| Counting |  |\n| --- | --- |\n| permutation P (n; r) combination 􀀀nr􀀁 Binomial theorem\u003Cbr>coins and pirates | number of ways to list r distinct elements from a set of size n number of ways to choose r elements from a set of size n\u003Cbr>n\u003Cbr>(x + y)n = P 􀀀nr􀀁 xryn􀀀r\u003Cbr>r=0\u003Cbr>number of ways to distribute r identical coins to n pirates: 􀀀r+nr􀀀1􀀁 |\n\n\n| Undirected graph G = (V; E)\u003Cbr>E 􀀒 ffu; vg j u 2 V ^ v 2 V ^ u  vg |\n| --- |\n| subgraph G0 = (V0 ; E0 ) where V0 􀀒 V and E0 􀀒 E walk v0 ; v 1 ; v2 ; : : : ; vn where fvi􀀀1; vig 2 E for all i\u003Cbr>trail walk with no repeated edges\u003Cbr>path walk with no repeated vertices\u003Cbr>cycle walk with no repeated vertices except v0 = v n\u003Cbr>connected contains a walk from any vertex to any other\u003Cbr>acyclic no subgraph is a cycle\u003Cbr>tree connected and acyclic =) jEj = j Vj 􀀀 1\u003Cbr>degree sum Pv2V deg(v) = 2jEj\u003Cbr>Eulerian contains a closed trail which visits every vertex\u003Cbr>clique set of vertices which are pairwise adjacent independent set set of vertices which are pairwise non-adjacent\u003Cbr>bip","cbCaiq6Lu4oGBjkZ","https://ap.wps.com/l/cbCaiq6Lu4oGBjkZ","pdf",88108,"English","# Set Theory Notation\n# Logic\n# Functions\n# Recursion and Master Theorem\n# Asymptotic Notation\n## Logarithm identities\n# Counting\n# Graph Theory","[{\"question\":\"What set theory symbols and operations are summarized in the cheat sheet?\",\"answer\":\"It lists common notation such as empty set, subset/proper subset, superset/proper superset, set equality, union, intersection, difference, symmetric difference, complement, Cartesian product, and power set, along with cardinality.\"},{\"question\":\"How are quantifiers and implications written in the logic section?\",\"answer\":\"It defines proposition and predicate logic, gives the notation for universal quantifier (for all, written 8) and existential quantifier (there exists, written 9), and shows implication and related forms like inverse, converse, and contrapositive.\"},{\"question\":\"Which key tools for recurrence relations and asymptotic growth are provided?\",\"answer\":\"It includes a recursion tree idea for T(n)=aT(n/b)+f(n) and the Master Theorem cases for comparing a and b^k, plus standard asymptotic notation (o, O, Θ, Ω, ω/! in the sheet) and their inequality-based definitions and combination rules.\"}]","Math 135 Cheat Sheet for Final Exam | PDF",1791926493]