Sets and Relations

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A set is a collection of well-defined objects. Objects are well-defined when we can clearly determine whether an element belongs to the collection or not. Well-defined examples: • Days in a week: {Mo

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**Roster Form:** A = {M, A, R, I, G, E} **Set-builder Form:** A = {x | x is a letter in the word MARRIAGE} **Note:** Even though 'A' appears twice in MARRIAGE, we write it only once in the set since

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**Definition:** An open interval (a,b) contains all real numbers between a and b, but excludes the endpoints a and b. **(2, 7) in set-builder form:** (2, 7) = {x | x ∈ R, 2 < x < 7} **Number line re

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**[3, 8) - Semi-closed interval:** [3, 8) = {x | x ∈ R, 3 ≤ x < 8} Includes 3, excludes 8 **(3, 8] - Semi-open interval:** (3, 8] = {x | x ∈ R, 3 < x ≤ 8} Excludes 3, includes 8 **[3, 8] - Closed in

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**Classifications:** • A = {} → Empty/Null set • B = {0} → Singleton set • C = {I, N, D, A} → Finite set • D = {1, 2, 3, 4, ...} → Infinite set **Cardinalities:** • n(A) = 0 (empty set has no element

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**Relationship:** A ⊂ B (A is a proper subset of B) - Every element of A is in B - B has at least one element (4) not in A - Therefore A ⊂ B and B ⊃ A **Equal vs Equivalent:** • **Equal:** A = B ✗ (t

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**Step-by-step solution:** **All subsets of A = {a, b}:** 1. {} (empty set) 2. {a} 3. {b} 4. {a, b} **Power set:** P(A) = {{}, {a}, {b}, {a, b}} **Number of elements:** n(P(A)) = 4 **Formula verif

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**Step 1: Find A'** A' = {x | x ∈ U and x ∉ A} A' = {1, 3, 5, 7} **Step 2: Find (A')'** (A')' = {x | x ∈ U and x ∉ A'} (A')' = {2, 4, 6, 8} **Step 3: Verification** (A')' = {2, 4, 6, 8} = A ✓ **Pro