1: SET AND LOGIC
- Page ID
- 202995
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets understand relationships between groups, and to analyze survey data.
- 1.1: Basics of Sets and Subsets
- This page covers the foundational concepts of sets in mathematics, including their definitions, notations, and the key contributions of Georg Cantor. It explains well-defined sets, cardinality, empty sets, and the differences between equal and equivalent sets, using practical examples. The page also clarifies subsets and proper subsets, along with their respective quantities calculated by \(2^n\) and \(2^n - 1\).
- 1.2: Set Operations- Union, Intersection, and Complement
- This page offers an overview of set theory focusing on union, intersection, and complement. It uses practical examples, including a comparison of sets from parents in the movie *Yours, Mine, and Ours*, to clarify concepts. Intersection represents common elements, while union combines all unique elements from both sets. The complement refers to elements outside a set within a universal context.
- 1.3: Cardinalty and Venn Diagrams with Survey Problems
- This page discusses the concept of cardinality in sets, representing it as |A| or n(A). It provides examples for calculating cardinality in unions and intersections, particularly in survey contexts, using Venn diagrams for clarity. The page highlights essential properties of cardinality, including formulas for unions and complements, with practical applications demonstrated through examples involving two and three sets to better understand survey results.
- 1.4: Statements, Compound Statements, and Negations
- This page covers logical statements and their evaluation as true or false, stressing the necessity of basing arguments on truth. It explains negation, quantifiers, and logical connectives, providing structures for affirmatives and their negations. The text addresses translating verbal statements into symbolic logic and discusses necessary and sufficient conditions, using examples and exercises to reinforce understanding of these concepts.
- 1.5: Truth Tables, Conditionals, and De Morgan’s Laws
- This page summarizes the key concepts of Boolean logic, including conjunction, disjunction, negation, and their truth tables. It covers conditional statements and their implications, as well as the converse, inverse, and contrapositive forms. The limitations of symbolic logic in interpreting English statements are noted, along with the introduction of De Morgan's Laws.
- 1.6: Analyzing Deductive Argument
- This page covers logical reasoning and argument validity, focusing on Venn and Euler diagrams for visualizing set relationships and their advantages over text. It explains various argument forms, including valid types like Modus Ponens, Modus Tollens, Disjunctive Syllogism, and the Transitive Property, while also detailing invalid arguments and common fallacies such as the Converse and Inverse.
- 1.7: Fallacies in Reasoning
- This page explains various logical fallacies that invalidate arguments. It covers types such as ad hominem, appeal to ignorance, appeal to authority, and false dilemma, along with circular reasoning, post hoc reasoning, and red herring. Each fallacy is illustrated with examples, highlighting how they compromise logical reasoning.
Thumbnail: Inclusion/exclusion for three sets. (CC BY-SA 3.0; unknown via Wikipedia).