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Math 214 — Comprehensive Exam Study Notes Study Notes

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Purpose: Short, actionable study plan and exam strategy tailored to Math 214 topics (Sets, Logic, Proofs). Use these notes as a checklist: understand definitions, work examples, practice short proofs, and memorize key equivalences and proof templates.

Study strategy:

Active practice: After reading a definition, do 2–3 short examples (membership, subset checks, truth tables, short proofs).
Proof templates: Keep one-page templates for each proof method: direct, contrapositive, proof by cases, vacuous/trivial. Practice converting statements between forms (e.g., implication ⇄ contrapositive).
Time management for exams: Spend first 5–10 minutes reading the whole paper. Do straightforward calculations and direct/proof-by-contrapositive problems first.

Example practice tasks to assign yourself:

From Sets: Given $S={1,2,3,4,5}$ , find $W={x\in S : x^2-6x+8=0}$ and compute $|\mathcal P(W)|$ .
From Logic: Construct the truth table for $P\Rightarrow(Q\land R)$ and verify De Morgan’s laws with a concrete $P,Q$ .
From Proofs: Prove “If $n$ is odd then $3n$ is odd” (direct), and “If $x^2$ is even then $x$ is even” (contrapositive template).

Tip: While practicing, explicitly label which method you use and why (e.g., “contrapositive is natural because showing $\neg Q\Rightarrow\neg P$ is easier”).

🧮 Chapter 1 — Sets (Key definitions & examples)

Definition — Set & Element: A set is a collection of objects called elements. We write membership as $a\in A$ and nonmembership as $a\notin A$ . The empty set is $\varnothing$ .

Notation & Examples:

List form: $X={a,b,c}$ . Order does not matter: ${1,2,3}={3,2,1}$ .
Set-builder: $W={x\in S : x^2-6x+8=0}$ . Example: if $S={1,2,3,4,5}$ then solve $x^2-6x+8=0$ to get $W={2,4}$ .

Special sets: $\mathbb R,\mathbb Q,\mathbb Z,\mathbb N$ and sign variants like $\mathbb R^+$ .

Cardinality: $|A|$ denotes the number of elements. If $|A|=n$ then $A$ is finite. Example: $A={1,2}$ has $|A|=2$ , $\varnothing$ has $|\varnothing|=0$ .

Subset & Equality: $A\subseteq B$ means every element of $A$ lies in $B$ . Two sets are equal iff each is a subset of the other (double inclusion). Proper subset: $A\subset B$ if $A\subseteq B$ and $A\neq B$ .

Power set: $\mathcal P(A)$ is the set of all subsets. If $|A|=n$ then $|\mathcal P(A)|=2^n$ . Example: for $A={1,2,3}$ list $\mathcal P(A)$ (eight subsets).

Set operations:

Union: $A\cup B={x: x\in A \text{ or } x\in B}$ .
Intersection: $A\cap B={x: x\in A \text{ and } x\in B}$ .
Difference: $A\setminus B={x: x\in A \text{ and } x\notin B}$ .
Complement: If universal set is $U$ , then $A^c={x\in U: x\notin A}$ .

Example: $S={a,b,c,d,e},\ T={d,e,f,g,h}$ . Then $S\cup T={a,b,c,d,e,f,g,h}$ and $S\cap T={d,e}$ .

Disjoint & Partition: Sets are disjoint if intersection is $\varnothing$ . A collection is a partition of $A$ if the union is $A$ and the pieces are pairwise disjoint (none empty).

Indexed unions/intersections: For $<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>e</mi><mi>t</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">sets</annotation></semantics></math>setsA_1,\dots,A_n<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mi>e</mi><mi>w</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">we write</annotation></semantics></math>wewrite\bigcup_{i=1}^n A_i<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>n</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">and</annotation></semantics></math>and\bigcap_{i=1}^n A_i<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo separator="true">;</mo><mi>m</mi><mi>o</mi><mi>r</mi><mi>e</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>l</mi><mi>l</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">; more generally</annotation></semantics></math>;moregenerally\bigcup_{i\in I}A_i<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mi>o</mi><mi>r</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>x</mi><mi>s</mi><mi>e</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">for index set</annotation></semantics></math>forindexsetI$ .

Cartesian product: $A\times B={(a,b):a\in A, b\in B}$ . Example: if $A={x,y}$ and $B={1,2,3}$ then $A\times B={(x,1),(x,2),(x,3),(y,1),(y,2),(y,3)}$ . Note $A\times B\neq B\times A$ in general.

Exam-style examples to practice:

Determine all subsets of $B={1,2,{3,4}}$ and compute $|B|$ .
Prove $|\mathcal P(A)|=2^{|A|}$ for small $A$ by listing and generalize the idea.
Given indexed sets $A_k=[k,k+1]$ for $k\in\mathbb Z$ , compute $\bigcup_{k\in\mathbb Z}A_k$ and $\bigcap_{k\in\mathbb Z}A_k$ .

🧠 Chapter 2 — Logic (Core concepts & quick examples)

Statement vs Open sentence: A statement is either true or false. An open sentence (predicate) like $P(x)$ becomes a statement when a value from the domain is substituted.

Logical connectives & truth behavior:

Negation: $\neg P$ (read: not $P$ ). Example: if $P$ is “Class begins at 9:30,” then $\neg P$ is “Class does not begin at 9:30.”
Disjunction: $P\lor Q$ is true if at least one of $P,Q$ is true.
Conjunction: $P\land Q$ is true only when both are true.
Implication (conditional): $P\Rightarrow Q$ (If $P$ then $Q$ ). It is only false when $P$ is true and $Q$ is false (vacuously true when $P$ is false).

Converse, Contrapositive, Biconditional:

Converse: reverse: $Q\Rightarrow P$ (not logically equivalent in general).
Contrapositive: $\neg Q\Rightarrow\neg P$ — logically equivalent to $P\Rightarrow Q$ and often useful for proofs.
Biconditional: $P\Leftrightarrow Q$ means both directions hold (true iff both have same truth value).

Logical equivalences & laws:

Commutative/associative/distributive laws hold for $\land,\lor$ (useful to simplify expressions).
De Morgan’s laws: $\neg(P\lor Q)\equiv (\neg P)\land(\neg Q)$ and $\neg(P\land Q)\equiv (\neg P)\lor(\neg Q)$ .

Tautology & Contradiction:

A tautology is always true (e.g., $P\lor\neg P$ ).
A contradiction is always false (e.g., $P\land\neg P$ ).

Quantifiers:

Universal: $\forall x\in S, P(x)$ — “for all $x$ in $S$ .”
Existential: $\exists x\in S, P(x)$ — “there exists $x$ in $S$ such that $P(x)$ .”
Negation rules: $\neg(\forall x, P(x)) \equiv \exists x, \neg P(x)$ and $\neg(\exists x, P(x)) \equiv \forall x, \neg P(x)$ .

Examples to practice:

Build the truth table for $P\Rightarrow(Q\land R)$ and check cases where it is false.
Use De Morgan to rewrite the negation of “ $P$ or $Q$ ”. Symbolically: $\neg(P\lor Q)\equiv(\neg P)\land(\neg Q)$ —practice with concrete $P,Q$ statements.
Quantifier example: Write “Every planet has a moon” as $\forall x\in\text{Planets},;P(x)$ . To disprove it produce a counterexample: $\exists x$ s.t. $\neg P(x)$ (e.g., Mercury).

Exam tips: Know when to use truth tables vs algebraic equivalences. For implications, always consider vacuous truth (premise false) and contrapositive proofs.

✍️ Chapter 3 — Proof techniques (Definitions, templates & examples)

Key vocabulary:

Axiom: accepted statement taken as true without proof.
Theorem/Proposition/Result: statement whose truth can be proved.
Lemma/Corollary: supporting result (lemma) or consequence (corollary) of a theorem.

Common proof methods (templates):

Direct proof: Assume hypothesis $P(x)$ for arbitrary $x\in S$ , deduce conclusion $Q(x)$ . Typical for algebraic manipulations.

Example: Prove "If $n<0$ and $n\in\mathbb Z$ , then $3<4$ ." (Trivial: the conclusion $3<4$ is always true.)
Concrete practice: Prove "If $n$ is odd then $3n$ is odd." Start: $n=2k+1$ , compute $3n=6k+3=2(3k)+1$ .

Proof by contrapositive: To prove $P\Rightarrow Q$ show $\neg Q\Rightarrow\neg P$ instead (logically equivalent).

Example: Prove "If $x^2$ is even then $x$ is even." Contrapositive: If $x$ is odd then $x^2$ is odd; show directly.

Proof by cases: Split into exhaustive cases (e.g., $n$ even or odd) and prove the claim in each case.

Example: Show $n^2+3n+7$ is odd for all $n\in\mathbb Z$ ; handle $n$ odd and $n$ even separately.

Vacuous & Trivial proofs:

Trivial: If conclusion is true for all elements, implication holds trivially.
Vacuous: If the hypothesis is false for all elements, the implication holds vacuously.

Useful integer facts (axioms / closure):

Integers closed under addition and multiplication; negatives are integers.
Definitions: even $n=2k$ , odd $n=2k+1$ , for $k\in\mathbb Z$ .
Example proofs using definitions: product of two evens is even; sum of two odds is even, etc.

Worked proof examples:

Example 1 (direct): If $n$ is odd, show $3n$ is odd. Proof sketch: Let $n=2k+1$ , then $3n=6k+3=2(3k)+1$ , so $3n$ is odd.
Example 2 (contrapositive): Show $x^2$ even $\Rightarrow x$ even. Contrapositive: If $x$ odd, write $x=2k+1$ and compute $x^2=4k^2+4k+1=2(2k^2+2k)+1$ , odd. Hence original implication holds.
Example 3 (cases): Prove $x+y$ even iff $x$ and $y$ have the same parity. Case 1: both even $\Rightarrow$ sum even; Case 2: both odd $\Rightarrow$ sum even. Converse by contraposition: if sum odd, parity differs.

Exam practice checklist:

For each statement: identify which proof method is most natural (direct, contrapositive, cases).
Write clear starts: "Let $x\in S$ . Suppose..." and explicit use of integer representation ( $2k$ , $2k+1$ ).
Close proofs with a concluding sentence like "Therefore the statement holds for all..." and optionally a Q.E.D. symbol.

Final tip: On the exam, annotate which method you use at the start. If stuck, try proving the contrapositive — often algebraic manipulations become simpler.

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Math 214 — Comprehensive Exam Study Notes Summary & Study Notes

📚 How to use these notes (from your request)

🧮 Chapter 1 — Sets (Key definitions & examples)

🧠 Chapter 2 — Logic (Core concepts & quick examples)

✍️ Chapter 3 — Proof techniques (Definitions, templates & examples)

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