Set Theory: Combined Operations

Combined operations involve the intersection, union and complement of sets. Perform the operations within brackets first. Other operations are performed from left to right.

Example:
Given that U = {x : 1 ≤ x ≤ 10, x is an integer},

G = {x : x is a prime number},

H = {x : x is an even number},

P = {1, 2, 3, 4, 5}.

List the elements of:

a) G ∩ H ∪ P

b) (G ∩ P) ’ ∪ H

c) H ’ ∩ (G ∪ P )

d) (P ∪ H ∪ G) ’ ∩ (G ∩ H)

Solution:
G = {2, 3, 5, 7}, H = {2, 4, 6, 8, 10}

a) G ∩ H ∪ P = {2} ∪ P ← G ∩ H = {2}

= {1, 2, 3, 4, 5}

b) (G ∩ P) ’ ∪ H = {1, 3, 4, 6, 7, 8, 9, 10} ∪ H

= {1, 2, 3, 4, 6, 7, 8, 9, 10}

c) H ’ ∩ (G ∪ P ) = H ’ ∩ {1, 2, 3, 4, 5, 7}

= {1, 3, 5, 7}

d) (P ∪ H ∪ G) ’ ∩ (G ∩ H) = {9} ∩ (G ∩ H)

= {9} ∩ {2} = { }

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