{"id":657,"date":"2021-08-05T13:38:31","date_gmt":"2021-08-05T13:38:31","guid":{"rendered":"https:\/\/sites.wcsu.edu\/roccac\/?page_id=657"},"modified":"2026-09-17T17:22:17","modified_gmt":"2026-09-17T17:22:17","slug":"foundational-discrete-mathematics","status":"publish","type":"page","link":"https:\/\/sites.wcsu.edu\/roccac\/homepage\/foundational-discrete-mathematics\/","title":{"rendered":"Foundational Discrete Mathematics"},"content":{"rendered":"\n<h1 class=\"wp-block-heading has-text-color\" style=\"color:#002856\"><strong><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2026\/08\/2026_Fall_MAT_141_Syllabus.pdf\" data-type=\"link\" data-id=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2024\/01\/2024_Spring_MAT_141_Discrete_Math_Syllabus.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Syllabus<\/a><\/strong><\/h1>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h1 class=\"wp-block-heading\"><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2026\/08\/MAT_141__Foundational_Discrete_Math_Notes.pdf\" data-type=\"link\" data-id=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/08\/MAT_141__Foundational_Discrete_Math_Notes.pdf\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Full Notes Packet<\/strong><\/a><\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If you find errors in the notes for MAT 141 Foundational Discrete Mathematics, then please fill out the following form: <a href=\"https:\/\/forms.office.com\/r\/y8vLJwZvSS\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/forms.office.com\/r\/y8vLJwZvSS<\/a>. Each error submitted can earn you an extra 0.5% on your grade up to +3% extra total.<\/li>\n\n\n\n<li>If you are comfortable DOING SO, you can instead create an &#8220;issue&#8221; on the GitHub repository where the code for the notes is stored: <a href=\"https:\/\/github.com\/cfroccajr\/MAT-141-Foundational-Discrete-Math-Notes\/issues\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/github.com\/cfroccajr\/MAT-141-Foundational-Discrete-Math-Notes\/issues<\/a><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-css-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading\"><span style=\"color:#002856\" class=\"tadv-color\">Class Calendar<\/span><\/h1>\n\n\n\n<p>There is a proposed calendar on the syllabus, but here I will record what we actually get through in each class.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n\n\n\n<li>9\/16\/2026: We discussed the first few examples of sequences.  We will pick up there on Friday.  Remember that you must turn in your practice exam at the start of class on Friday, if you are typing it make sure to print it out.  Late practice exams will get at most a 75%. We will spend 15-20 minutes answering questions about the practice exam before continuing with the new material.<\/li>\n\n\n\n<li>9\/14\/2026: We finished off the last of the notes on quantifiers and predicates.  You now have all you need to complete the practice exam for unit 1 which is due Friday.  The exam is next Monday as it says on the syllabus.<\/li>\n\n\n\n<li>9\/11\/2026: We looked at additional examples of predicates and quantified statements.  We will finish off chapter 3 on schedule on Monday and perhaps start a little of the next unit.<\/li>\n\n\n\n<li>9\/9\/2026: We did another example of combining logical arguments to create larger arguments and then we introduced predicates. We discussed different ways we can &#8220;assign&#8221; values to variables in predicates, the effect of quantifiers, and how to negate quantifiers.<\/li>\n\n\n\n<li>9\/4\/2026: We discussed valid argument forms, critical rows, and how to combine our basic valid arguments to make longer arguments.  Please try and complete practice problem 1.15 (p.18) for Wednesday so that we can go over it quickly and start on the next section.  If you didn&#8217;t turn in your work today, then you need to get it to me by Wednesday if you don&#8217;t want it counted late, and you must adhere to the guidelines from the syllabus that we discussed in class.<\/li>\n\n\n\n<li>9\/2\/2026: Today we discussed implications, converses, inverses, contrapositives, and negations.  We also looked at wheat we mean when we say something is necessary versus sufficient (it is sufficient to be a multiple of 6 in order to be a multiple of 3, it is necessary to be a multiple of 3 in order to be a multiple of 6)  Me also touched on some logical argument forms, we will pick up there next time.<\/li>\n\n\n\n<li>8\/31\/2026: Today we started in the printed notes packet. We covered section 1.1 and 1.2 in the notes, these correspond to the start of Chapter 2 in the text. If you didn&#8217;t bring your binder today, bring it Wednesday so that you can get the notes packet.  Also, recall that the <a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/01\/Writing_Up_Mathematics_Assessment_Tool-1.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Writing Up Mathematics Assignment<\/a> is due on Friday.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading\"><span style=\"color:#002856\" class=\"tadv-color\">Unit 1: Logic<\/span><\/h1>\n\n\n\n<h4 class=\"wp-block-heading has-text-color has-link-color wp-elements-45a931920409da5ffc9e66002f89aad2\" style=\"color:#002856\"><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/01\/Writing_Up_Mathematics_Assessment_Tool-1.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Writing Up Mathematics Assignment Due 9\/4<\/a><\/h4>\n\n\n\n<h4 class=\"wp-block-heading has-text-color\" style=\"color:#002856\"><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/01\/MAT_141_Unit_I_Study_Guide__Spring_2025_Edition_.pdf\" data-type=\"link\" data-id=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/01\/MAT_141_Unit_I_Study_Guide__Spring_2025_Edition_.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Practice Exam for Unit 1 Due 9\/18<\/a><\/h4>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading has-text-color\" style=\"color:#002856\">Unit 2: Sequences and Summations<\/h1>\n\n\n\n<h4 class=\"wp-block-heading has-text-color\" style=\"color:#002856\"><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/03\/MAT_141_Unit_II_Study_Guide__Spring_2025_.pdf\" data-type=\"link\" data-id=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/03\/MAT_141_Unit_II_Study_Guide__Spring_2025_.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Practice Exam for Unit 2 Due 10\/9<\/a><\/h4>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading has-text-color\" style=\"color:#002856\">Unit 3: Sets, Functions, Relations<\/h1>\n\n\n\n<h4 class=\"wp-block-heading has-text-color\" style=\"color:#002856\"><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2026\/01\/2025_Spring_MAT_141_Unit_III_Study_Guide.pdf\" data-type=\"link\" data-id=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/03\/2025_Spring_MAT_141_Unit_III_Study_Guide.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Practice Exam for Unit 3 Due 10\/30<\/a><\/h4>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\" \/>\n\n\n\n<p>Extra Credit Exercise: The code below will build the equivalence classes from #26 in sec. 8.2. Explain how the code works both practically and based on the definition of the relation in 26.<\/p>\n\n\n\n<pre class=\"wp-block-code\" style=\"font-size:13px\"><code># Written in Python\n# Empty collection of equivalence classes\nEC={}\n# Build equivalence classes\nfor i in range(3):\n    for j in range(3):\n        for k in range(3):\n            for l in range(3):\n                # New element\n                temp_lst=&#091;i,j,k,l]\n                # Assign to equivalence class based on sum\n                EC&#091;sum(temp_lst)]=EC.get(sum(temp_lst),&#091;])+&#091;temp_lst]\n# Display equivalence classes\nfor key in EC: print(key,\":\\t\",EC&#091;key])<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-css-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading has-text-color\" style=\"color:#002856\">Unit 4: Counting, Combinatorics, and Graphs<\/h1>\n\n\n\n<h4 class=\"wp-block-heading has-text-color\" style=\"color:#002856\"><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2026\/01\/MAT_141_Unit_IV_Study_Guide__Spring_2025_.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Practice Exam for Unit 4 Due 12\/4<\/a><\/h4>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading has-text-color\" style=\"color:#002856\">Extra Credit:<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"tadv-color\" style=\"color:#ee4d00\">Typesetting Out-of-Class Work in \\(\\LaTeX\\) (+5% on each typed up piece of work)<\/span><\/h2>\n\n\n\n<p>If you are doing this extra credit here is a quick reference sheet with symbols and such: <a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2023\/01\/LaTeX_Quick_Reference.pdf\" target=\"_blank\" rel=\"noreferrer noopener\"><em><span class=\"tadv-color\" style=\"color:#ee4d00\"><strong>Quick Reference<\/strong><\/span><\/em><\/a><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><span class=\"tadv-color\" style=\"color:#002856\"><em>Create an account at <a href=\"http:\/\/overleaf.com\" target=\"_blank\" rel=\"noreferrer noopener\">Overleaf.com<\/a> using your WCSU email account<\/em><\/span><\/li>\n\n\n\n<li>Watch the introductory video here <a href=\"https:\/\/media.wcsu.edu\/media\/Using+the+Homework+Template\/1_ij1s2qy2\" target=\"_blank\" rel=\"noreferrer noopener\"><strong><em><span style=\"color:#ee4d00\" class=\"tadv-color\">VIDEO LINK<\/span><\/em><\/strong><\/a>, the document created in the video is here: <a rel=\"noreferrer noopener\" href=\"https:\/\/www.overleaf.com\/read\/khbktbvvvkjd\" target=\"_blank\"><strong><em><span style=\"color:#ee4d00\" class=\"tadv-color\">https:\/\/www.overleaf.com\/read\/khbktbvvvkjd<\/span><\/em><\/strong><\/a><\/li>\n\n\n\n<li><span class=\"tadv-color\" style=\"color:#002856\">Follow this link to see samples of typed up exercises: <\/span><a href=\"https:\/\/www.overleaf.com\/latex\/templates\/homework-template-with-samples\/zmmdzbkzvtzk\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/www.overleaf.com\/latex\/templates\/homework-template-with-samples\/zmmdzbkzvtzk<\/a><\/li>\n\n\n\n<li><span style=\"color:#002856\" class=\"tadv-color\">Follow this link to make a copy of a blank template:<\/span> <a rel=\"noreferrer noopener\" href=\"https:\/\/www.overleaf.com\/read\/hwdhcvmntrvs\" target=\"_blank\"><span style=\"color:#ee4d00\" class=\"tadv-color\"><strong><em>https:\/\/www.overleaf.com\/read\/hwdhcvmntrvs<\/em><\/strong><\/span><\/a><\/li>\n\n\n\n<li><span style=\"color:#002856\" class=\"tadv-color\">Submit work by downloading a copy of the PDF you generate and printing it.<\/span><\/li>\n<\/ol>\n\n\n\n<p>If you are having problems typing up a document and would like feedback you can share a link to it by following the directions here: <a href=\"https:\/\/www.overleaf.com\/learn\/how-to\/Sharing_a_project\" target=\"_blank\" rel=\"noreferrer noopener\"><strong><em>https:\/\/www.overleaf.com\/learn\/how-to\/Sharing_a_project<\/em><\/strong><\/a>; send me the link to view but not edit.  You can see a full playlist of video lessons here: <a href=\"https:\/\/media.wcsu.edu\/playlist\/dedicated\/1_m067qwdq\/\" target=\"_blank\" rel=\"noreferrer noopener\">Technical Typesetting with \\(LaTeX\\) Playlist<\/a><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\" \/>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Additional Extra Credit:<\/strong> <\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Complete this <a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/10\/Reflecting_on_Quantifiers.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Reflection on Quantifiers<\/a> for upto +2% on your final grade.<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading\"><span style=\"color:#002856\" class=\"tadv-color\">Links and Handouts<\/span><\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a rel=\"noopener\" href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2021\/08\/Set-Laws.pdf\" target=\"_blank\">Sets Reference Table<\/a><\/li>\n\n\n\n<li><a rel=\"noopener\" href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2021\/08\/PotsdamTables.pdf\" target=\"_blank\">Logic Reference Table<\/a><\/li>\n\n\n\n<li><a rel=\"noopener\" href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2021\/09\/Blank_Truth_Tables.pdf\" target=\"_blank\">Blank Truth Tables<\/a><\/li>\n\n\n\n<li><a rel=\"noopener\" href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2021\/08\/ProbabilityNotes.pdf\" target=\"_blank\">Probability Notes<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2020\/06\/Writing-Math-Well-HMC-Math.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">\u201cWriting Mathematics Well\u201d<\/a>&nbsp;and&nbsp;<a href=\"https:\/\/math.hmc.edu\/su\/wp-content\/uploads\/sites\/10\/2020\/08\/Guidelines-for-Good-Mathematical-Writing.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">\u201cGuidelines for Good Mathematical Writing\u201d<\/a>&nbsp;by Francis Su<\/li>\n\n\n\n<li><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2025\/01\/Writing_Up_Mathematics_Assessment_Tool-1.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Writing up Mathematics Packet<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2026\/08\/Presentation___Writing_Up_a_Math_Problem.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Writing up Mathematics Presentation<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/media.wcsu.edu\/playlist\/dedicated\/1_m067qwdq\/\" target=\"_blank\" rel=\"noreferrer noopener\">Technical Typesetting with \\(LaTeX\\) Playlist<\/a><\/li>\n\n\n\n<li><a href=\"http:\/\/overleaf.com\/\" target=\"_blank\" rel=\"noreferrer noopener\">Overleaf: LaTeX Document Preparation<\/a><\/li>\n\n\n\n<li><a rel=\"noreferrer noopener\" href=\"https:\/\/www.youtube.com\/c\/WritelatexOnline\/videos\" target=\"_blank\">Overleaf Tutorial Videos<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/media.wcsu.edu\/playlist\/dedicated\/1_m067qwdq\/\" target=\"_blank\" rel=\"noreferrer noopener\"><\/a><a href=\"http:\/\/www.utc.fr\/~jlaforet\/Suppl\/latex-cheatsheet.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">LaTeX Quick Reference<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2023\/01\/LaTeX_Quick_Reference.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">LaTeX Quick Reference v.2<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.overleaf.com\/latex\/templates\/homework-template-with-samples\/zmmdzbkzvtzk\" target=\"_blank\" rel=\"noreferrer noopener\">Homework Template with Samples<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.overleaf.com\/read\/hwdhcvmntrvs\" target=\"_blank\" rel=\"noreferrer noopener\">Homework Template Blank<\/a><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-css-opacity is-style-wide\" \/>\n\n\n\n<h1 class=\"wp-block-heading\"><span style=\"color:#002856\" class=\"tadv-color\">Vocabulary<\/span><\/h1>\n\n\n\n<p class=\"has-text-color\" style=\"color:#002658\">\\(\\mathbb{N}\\),&nbsp;\\(\\mathbb{Z}\\),&nbsp;\\(\\mathbb{Q}\\),&nbsp;\\(\\mathbb{R}\\), rational number, irrational number, set notation, set roster notation, modus ponens, modus tollens, De Morgan&#8217;s Law, negation, distribution, commutative law, associative law, \\(\\forall\\), \\(\\exists\\), converse, inverse, contrapositive, conjunction, disjunction,&nbsp; set, subset, Cartesian product, relation, function, sequences, series, summation, summation notation \\[\\sum_{i=0}^n a_i=a_0+a_1+\\cdots+a_n,\\] product notation \\[\\prod_{i=0}^n a_i=a_0\\times a_1\\times \\cdots\\times a_n,\\] geometric sum \\[\\sum_{i=0}^n a\\, r^i=a+ar+ar^2+\\cdots+ar^n=a\\, \\frac{(r^{n+1}-1)}{r-1},\\] union, intersection, set difference, set complement, power set &#8211; \\(\\mathscr{P}(x)\\), reflexive, symmetric, transitive, anti-symmetric, function, one-to-one, onto, inverses, composition, addition principle, multiplication principle, possibilities tree, combinations, factorial, permutations, binomial coefficients, multinomial coefficients, graph, walk, circuit, Euler circuit, path, tree, trail, binary tree, Hamilton circuit, closed walk, \u2026 etc.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Syllabus Full Notes Packet Class Calendar There is a proposed calendar on the syllabus, but here I will record what we actually get through in each class. Unit 1: Logic Writing Up Mathematics Assignment Due 9\/4 Practice Exam for Unit &hellip; <a href=\"https:\/\/sites.wcsu.edu\/roccac\/homepage\/foundational-discrete-mathematics\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":35,"featured_media":0,"parent":2,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"wpo365_audiences":[],"wpo365_private":false,"footnotes":""},"class_list":["post-657","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages\/657","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/users\/35"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/comments?post=657"}],"version-history":[{"count":51,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages\/657\/revisions"}],"predecessor-version":[{"id":2328,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages\/657\/revisions\/2328"}],"up":[{"embeddable":true,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages\/2"}],"wp:attachment":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/media?parent=657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}