Set Theory → Union and Intersection (Intermediate)

1) Given A = {1, 2, 3}, B = {3, 4}, find (A ∪ B)′ if U = {1, 2, 3, 4, 5}.

{5}

{1}

{2, 5}

{3, 4}

A ∪ B = {1, 2, 3, 4}; complement in U = {5}.

Explanation

A ∪ B = {1, 2, 3, 4}; complement in U = {5}.

2) Given A = {x, y}, B = {y, z}, find (A ∩ B)′ if U = {w, x, y, z}.

{w, x, z}

{y}

{w, y, z}

{x, y}

A ∩ B = {y}; complement = {w, x, z}.

Explanation

A ∩ B = {y}; complement = {w, x, z}.

3) Given A = {1, 3, 5}, B = {2, 4, 6}, C = {3, 6}, find (A ∪ B) ∩ C.

{3, 6}

{2, 3}

{6}

{1, 5}

A ∪ B = {1, 2, 3, 4, 5, 6}; intersect with C = {3, 6}.

Explanation

A ∪ B = {1, 2, 3, 4, 5, 6}; intersect with C = {3, 6}.

4) Given A = {2, 4, 6}, B = {4, 6, 8}, find (A′ ∩ B′) if U = {2, 4, 6, 8, 10}.

{10}

{8}

{2}

{6, 10}

A′ = {8, 10}, B′ = {2, 10}; intersection = {10}.

Explanation

A′ = {8, 10}, B′ = {2, 10}; intersection = {10}.

5) Given A = {1, 2}, B = {2, 3}, C = {3, 4}, find A ∪ (B ∩ C).

{1, 2, 3}

{1, 2}

{2, 3}

{1, 3, 4}

B ∩ C = {3}; union with A = {1, 2, 3}.

Explanation

B ∩ C = {3}; union with A = {1, 2, 3}.

6) Given A = {a, b}, B = {b, c}, C = {c, d}, find (A ∪ B) ∩ C.

{c}

{b, c}

{d}

{a, c}

A ∪ B = {a, b, c}; intersect with C = {c}.

Explanation

A ∪ B = {a, b, c}; intersect with C = {c}.

7) Given A = {1, 2, 3}, B = {2, 3, 4}, find A′ ∪ B if U = {1, 2, 3, 4, 5}.

{2, 3, 4, 5}

{1, 2, 3}

{3, 4, 5}

{1, 4, 5}

A′ = {4, 5}; union with B = {2, 3, 4, 5}.

Explanation

A′ = {4, 5}; union with B = {2, 3, 4, 5}.

8) Given A = {p, q}, B = {q, r}, find (A ∪ B)′ ∪ B if U = {p, q, r, s}.

{q, r, s}

{p, q, r}

{q, s}

{p, s}

A ∪ B = {p, q, r}; complement = {s}; union with B = {q, r, s}.

Explanation

A ∪ B = {p, q, r}; complement = {s}; union with B = {q, r, s}.

9) Given A = {1, 4, 5}, B = {2, 4, 6}, find (A ∩ B)′.

{1, 2, 5, 6}

{4}

{1, 5}

{2, 6}

A ∩ B = {4}; complement in U = {1, 2, 5, 6}.

Explanation

A ∩ B = {4}; complement in U = {1, 2, 5, 6}.

10) Given A = {x, y}, B = {y, z}, C = {z, w}, find (A ∪ B ∪ C)′ if U = {v, w, x, y, z}.

{v}

{w}

{z}

∅

A ∪ B ∪ C = {w, x, y, z}; complement in U = {v}.

Explanation

A ∪ B ∪ C = {w, x, y, z}; complement in U = {v}.