# If A = {1, 2, 3}, B = {3, 4}

Question:

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(i) A × (B ∩ C)

(ii) (A × B) ∩ (A × C)

(iii) A × (B ∪ C)

(iv) (A × B) ∪ (A × C)

Solution:

Given:

A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}

(i) A × (B ∩ C)

Now,

(B ∩ C)  = {4}

∴ A × (B ∩ C)  = {(1, 4), (2, 4), (3, 4)}

(ii) (A × B) ∩ (A × C)

Now,

(A × B) = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}

And,

(A × C) = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}

∴ (A × B) ∩ (A × C) = {(1, 4), (2, 4), (3, 4)}

(iii) A × (B ∪ C)

Now,

(B ∪ C) = {3, 4, 5, 6}

∴ A × (B ∪ C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}

(iv) (A × B) ∪ (A × C)

Now,

(A × B) = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}

And,

(A × C) = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}

∴ (A × B) ∪ (A × C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}