Sets, Relations, and Functions Cheatsheet

Cheat Codes & Shortcuts

• Set: A collection of distinct objects, e.g., \( A = \{1, 2, 3\} \).
• Union: \( A \cup B = \{ x \mid x \in A \text{ or } x \in B \} \).
• Intersection: \( A \cap B = \{ x \mid x \in A \text{ and } x \in B \} \).
• Complement: \( A' = \{ x \in U \mid x \notin A \} \), where \( U \) is the universal set.
• Relation: A subset of Cartesian product \( A \times B \), e.g., \( R = \{(a, b) \mid a \in A, b \in B\} \).
• Function: A relation from \( A \) to \( B \) where each \( a \in A \) maps to exactly one \( b \in B \).
• Injective: \( f(a_1) = f(a_2) \implies a_1 = a_2 \).
• Surjective: For every \( b \in B \), there exists \( a \in A \) such that \( f(a) = b \).
• Bijective: Both injective and surjective.
• Inverse Function: If \( f: A \to B \) is bijective, then \( f^{-1}: B \to A \).

Quick Reference Table

Type	Form	Example/Method
Union	\( A = \{1, 2\}, B = \{2, 3\} \)	\( A \cup B = \{1, 2, 3\} \)
Intersection	\( A = \{1, 2\}, B = \{2, 3\} \)	\( A \cap B = \{2\} \)
Relation	\( R \subseteq A \times B \)	\( R = \{(1, a), (2, b)\} \)
Function	\( f: A \to B \)	\( f(x) = x^2 \), maps \( x \) to \( x^2 \)
Injective	\( f(x) = 2x \)	Check \( f(x_1) = f(x_2) \implies x_1 = x_2 \)
Surjective	\( f: \mathbb{R} \to \mathbb{R}, f(x) = x \)	Every \( y \in \mathbb{R} \) has \( x = y \)

Advice

Sets, Relations, and Functions Quick Tips

• Set Operations: Use \( \cup \), \( \cap \), and \( ' \) for union, intersection, and complement.
• Relation Properties: Check for reflexivity (\( (a, a) \in R \)), symmetry (\( (a, b) \in R \implies (b, a) \in R \)), transitivity (\( (a, b), (b, c) \in R \implies (a, c) \in R \)).
• Function Check: Ensure each input maps to exactly one output.
• Injective Test: Solve \( f(x_1) = f(x_2) \) to check if \( x_1 = x_2 \).
• Surjective Test: Confirm every element in codomain has a preimage.

Sets, Relations, and Functions Speed Quiz

Test your speed with 5 sets, relations, and functions questions! You have 30 seconds per question.