Sets

{1, 3, 5}, ∅, {1, 2, 3, 4, 5}, {3}

A subset contains only elements that are also in the original set. {1, 3, 5} ✓ - all elements are in A. ∅ ✓ - empty set is subset of every set. {1, 2, 3, 4, 5} ✓ - every set is subset of itself. {6} ✗ - 6 is not in A. {2, 4, 6} ✗ - 6 is not in A. {3} ✓ - 3 is in A.

{x : x = 2n, n ∈ N, n ≤ 5}

Step 1: Identify the pattern: 2, 4, 6, 8, 10 are first 5 positive even numbers. Step 2: Even numbers can be written as 2n where n is a natural number. Step 3: For first 5 even numbers: n ∈ {1, 2, 3, 4, 5}, so n ≤ 5. Therefore: {x : x = 2n, n ∈ N, n ≤ 5}.

8

Step 1: For a set with n elements, number of subsets = 2ⁿ. Step 2: Here n = 3, so number of subsets = 2³ = 8. Step 3: The subsets are: ∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}. Each element can either be included or excluded, giving 2 choices per element.

∅, {}, {x : x ∈ N, x < 1}, {x : x² = -1, x ∈ R}

∅ ✓ - standard symbol for empty set. {} ✓ - empty braces represent empty set. {x : x ∈ N, x < 1} ✓ - no natural numbers less than 1. {0} ✗ - contains element 0, not empty. {x : x² = -1, x ∈ R} ✓ - no real number squared equals -1.