Chapter Review

Number Systems, Sets, and Functions

Complex Numbers · Sets and Logic · Functions and Groups

Complex Numbers and the Imaginary Unit

Complex numbers extend the reals by introducing $i = \sqrt{-1}$, written in standard form $z = a + bi$ where $a$ is the real part and $b$ is the imaginary part.

Key Points

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Powers of $i$ cycle every 4: $i, -1, -i, 1$ — use $i^n = i^{n \bmod 4}$
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Two complex numbers are equal iff their real and imaginary parts match separately
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Every real number is a complex number with imaginary part 0
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Any four consecutive powers of $i$ sum to zero

Formula

$$i^2 = -1$$

Conjugate, Modulus, and Division

The conjugate $\bar{z} = a - bi$ reflects across the real axis. Multiplying $z$ by its conjugate gives the square of its modulus, which is central to complex division.

Key Points

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$z\bar{z} = a^2 + b^2 = |z|^2$ — always a non-negative real number
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Modulus: $|z| = \sqrt{a^2 + b^2}$ (distance from origin on the Argand diagram)
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Division: multiply numerator and denominator by the conjugate of the denominator
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Conjugate distributes over +, ×, and ÷: $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$
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Modulus of a product equals product of moduli: $|z_1 z_2| = |z_1||z_2|$
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Triangle inequality: $|z_1 + z_2| \le |z_1| + |z_2|$

Formula

$$z\bar{z} = |z|^2 = a^2 + b^2$$

Polar Form and De Moivre's Theorem

Polar form $z = r(\cos\theta + i\sin\theta)$ uses modulus $r$ and argument $\theta$, simplifying multiplication and exponentiation of complex numbers.

Key Points

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Conversion: $r = \sqrt{x^2 + y^2}$, $\theta = \tan^{-1}(y/x)$ with quadrant adjustment
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Multiplication in polar: multiply moduli, add arguments
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De Moivre's Theorem: $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$
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$\tan^{-1}(y/x)$ alone only gives correct angle in Q1 and Q4 — add $\pi$ for Q2/Q3

Formula

$$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$$

Set Fundamentals and Operations

A set is a well-defined collection of distinct elements. Union, intersection, and complement are the core operations, visualized through Venn diagrams.

Key Points

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Union ($A \cup B$): all elements in either set — corresponds to logical OR
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Intersection ($A \cap B$): only elements in both sets — corresponds to logical AND
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Complement ($A'$): everything in $U$ not in $A$; depends on the universal set
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Set difference: $A - B = A \cap B'$
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Disjoint sets have empty intersection: $A \cap B = \emptyset$

Formula

$$|A \cup B| = |A| + |B| - |A \cap B|$$

Set Algebra and De Morgan's Laws

Set operations follow algebraic laws (commutative, associative, distributive). De Morgan's Laws connect complement with union/intersection by flipping the operation.

Key Points

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Commutative: $A \cup B = B \cup A$, $A \cap B = B \cap A$
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Distributive: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ — works both ways for sets
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De Morgan's: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$
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Complement laws: $A \cup A' = U$ and $A \cap A' = \emptyset$
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Power set cardinality: $|P(S)| = 2^n$ for a set with $n$ elements

Formula

$$(A \cup B)' = A' \cap B'$$

Propositional Logic and Connectives

Propositions are statements with definite truth values combined using conjunction (∧), disjunction (∨), and negation (¬) to build compound statements evaluated via truth tables.

Key Points

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Conjunction ($p \land q$): true only when both are true
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Disjunction ($p \lor q$): false only when both are false (inclusive OR)
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Negation ($\neg p$): flips truth value; double negation cancels
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Tautology: always true (e.g., $p \lor \neg p$); Contradiction: always false (e.g., $p \land \neg p$)
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Truth table has $2^n$ rows for $n$ variables

Conditional and Biconditional Statements

The conditional $p \to q$ is false only when $p$ is true and $q$ is false. The contrapositive $\neg q \to \neg p$ is logically equivalent; the converse $q \to p$ is not.

Key Points

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$p \to q$ is equivalent to $\neg p \lor q$
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Contrapositive ($\neg q \to \neg p$) is equivalent to the original — valid for proofs
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Converse ($q \to p$) and inverse ($\neg p \to \neg q$) are NOT equivalent to the original
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Biconditional ($p \leftrightarrow q$): true when both have the same truth value
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A false hypothesis makes any conditional vacuously true

Formula

$$p \to q \equiv \neg p \lor q$$

Types of Functions: Injective, Surjective, Bijective

Functions are classified by how they pair domain and codomain elements. Only bijective functions (both one-to-one and onto) have inverses.

Key Points

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Injective (one-to-one): distinct inputs always give distinct outputs — passes the horizontal line test
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Surjective (onto): every codomain element is mapped to by at least one input — range equals codomain
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Bijective: both injective and surjective — a perfect one-to-one correspondence
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Only bijective functions have well-defined inverse functions

Composition and Inverse Functions

Composition $(g \circ f)(x) = g(f(x))$ chains functions by feeding output of $f$ into $g$. An inverse function reverses the mapping: $f^{-1}(f(x)) = x$.

Key Points

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Composition is NOT commutative: $g \circ f \neq f \circ g$ in general
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To find $f^{-1}$: set $y = f(x)$, swap $x$ and $y$, solve for $y$
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Graph of $f^{-1}$ is the reflection of $f$ across the line $y = x$
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Composing with inverse gives identity: $f^{-1}(f(x)) = x$

Formula

$$(g \circ f)(x) = g(f(x))$$

Groups and Algebraic Structures

A group $(G, *)$ is a set with a binary operation satisfying closure, associativity, identity, and inverse. Adding commutativity makes it an Abelian group.

Key Points

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Four axioms: closure, associativity, identity element $e$, and inverse $a^{-1}$ for every element
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Abelian group: additionally satisfies $a * b = b * a$ (commutative)
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Hierarchy: Groupoid (closure) → Semi-group (+associativity) → Monoid (+identity) → Group (+inverse)
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Reversal law: $(a * b)^{-1} = b^{-1} * a^{-1}$
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Cayley table symmetric about diagonal ⟹ Abelian

Formula

$$a * a^{-1} = a^{-1} * a = e$$

Formulas

Complex Modulus

Distance of a complex number from the origin on the Argand diagram.

Formula

$|z| = \sqrt{a^2 + b^2}$

Conjugate Product

Product of a complex number with its conjugate equals the square of its modulus.

Formula

$z\bar{z} = a^2 + b^2 = |z|^2$

De Moivre's Theorem

Raises a complex number in polar form to an integer power.

Formula

$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$

Inclusion-Exclusion Principle

Counts union size by adding individual sets and subtracting overlap.

Formula

$|A \cup B| = |A| + |B| - |A \cap B|$

De Morgan's Laws (Sets)

Complement of union = intersection of complements, and vice versa.

Formula

$(A \cup B)' = A' \cap B'$

Conditional Equivalence

Converts an if-then statement into a disjunction.

Formula

$p \to q \equiv \neg p \lor q$

Composite Function

Applying function g to the result of function f.

Formula

$(g \circ f)(x) = g(f(x))$

Group Inverse Property

Every group element has an inverse that returns to the identity.

Formula

$a * a^{-1} = a^{-1} * a = e$