Question

Verify n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) – n(A $\cap$ B) – n(B $\cap$ C) – n(A $\cap$ C) + n(A $\cap$ B $\cup$ C) for the following sets. A = {a, c, e, f, h}, B = {c, d, e, f} and C = {a, b, c, f}

Answer 1

n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) – n(A $\cap$ B) – n(B $\cap$ C) – n(A $\cap$ C) + n(A $\cap$ B $\cap$ C)

A = {a, c, e, f, h}, B = {c, d, e, f}, C = {a, b, c, f}

n (A) = 5, n (B) = 4, n (C) = 4

n( A $\cap$ B) =3

n(B $\cap$ C) = 2

n( A $\cap$ C) =3

n( A $\cap$ B $\cap$ C) = 2

A $\cap$ B = {c, e, f}

B $\cap$ C = {c, f}

A $\cap$ C = {a, c, f}

A $\cap$ B $\cap$ C = {c, f}

A $\cup$ B $\cup$ C = {a, c, d, e, f, b, h}

$\therefore$ n(A $\cup$ B $\cup$ C) = 7 ……………. (1)

n(A) + n(B) + n(C) – n(A $\cap$ B) – n(B $\cap$ C) – n(A $\cap$ C) + n(A $\cap$ B $\cap$ C) = 5 + 4 + 4 – 3 – 2 – 3 + 2 = 15 – 8 = 7 …(2)

$\therefore$ (1) =(2)

$\Rightarrow$ n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) – n(A $\cap$ B) – n(B $\cap$ C) – n(A $\cap$ C) + n(A $\cap$ B $\cap$ C)

Hence it is verified.