Question

If A = `{y : y = ("a"+1)/2, "a" ∈ "W" and "a" ≤ 5}`, B = `{y : y = (2"n" – 1)/2, "n" ∈ "W" and "n" < 5}` and C = `{-1, −1/2, 1, 3/2, 2}` then show that A – (B ∪ C) = (A – B) ∩ (A – C)

Answer 1

A = `{y : y = ("a"+1)/2, "a" ∈ "W" and "a" ≤ 5}`

a = {0, 1, 2, 3, 4, 5}

⇒ y = `("a" + 1)/2 = 1/2`

y = `(1 + 1)/2 = 2/2` = 1

y = `(2 + 1)/2 = 3/2`

y = `(3 + 1)/2 = 4/2` = 2

y = `(4 + 1)/2 = 5/2`

y = `(5 + 1)/2 = 6/2` = 3

∴ A = `{1/2, 1, 3/2, 2, 5/2, 3}`

B = `{y : y = (2"n" – 1)/2, "n" ∈ "W" and "n" < 5}`

n = {0, 1, 2, 3, 4}

⇒ y = `(2 xx 0 - 1)/2 = (-1)/2`

y = `(2 xx 1 - 1)/2 = 1/2`

y = `(2 xx 2 - 1)/2 = 3/2`

y = `(2 xx 3 - 1)/2 = 5/2`

y = `(2 xx 4 - 1)/2 = 7/2`

∴ B = `{-1/2, 1/2, 3/2, 5/2, 7/2}`

C = `{-1, -1/2, 1, 3/2, 2}`

B ∪ C = `{-1, -1/2, 1/2, 1, 3/2, 2, 5/2, 7/2}`

A − (B ∪ C) = {3}    ...(1)

A − B = {1, 2, 3}

A − C = `{1/2, 5/2, 3}`

(A – B) ∩ (A – C) = {3}  ...(2)

From (1) and (2), it is verified that

A – (B ∪ C) = (A – B) ∩ (A – C).