What is a Set? Explain types of sets and set operations with examples.

List all elements inside curly braces. Example: Set of vowels = {a, e, i, o, u}

Describe the property elements must satisfy. Example: A = {x | x is a natural number less than 5} = {1, 2, 3, 4}

A set with no elements. Example: {x | x² = -1, x is real}

A set with a countable number of elements. Example: {1, 2, 3, 4, 5}

A set with unlimited elements. Example: Set of all natural numbers N = {1, 2, 3, ...}

The set containing all objects under consideration in a problem.

A is a subset of B (A ⊆ B) if every element of A is also in B. Example: {1,2} ⊆ {1,2,3}

The set of all subsets of A. If |A| = n, then |P(A)| = 2ⁿ. Example: A = {1,2} → P(A) = {∅, {1}, {2}, {1,2}}

Two sets A and B are equal if they contain exactly the same elements: A = B iff A ⊆ B and B ⊆ A

All elements in A OR B (or both). A ∪ B = {x | x ∈ A or x ∈ B}

Elements in BOTH A AND B. A ∩ B = {x | x ∈ A and x ∈ B}

Elements in A but NOT in B. A − B = {x | x ∈ A and x ∉ B}

All elements in U that are NOT in A. A' = U − A

Elements in A or B but NOT in both. A △ B = (A − B) ∪ (B − A)

(A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∪ A = A and A ∩ A = A

A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A