Verify n(A ∪ B ∪ C) = n(A) + n(B) + n(C) –n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) for the sets given below:

A = {a,b,c,d,e}, B = {x,y,z} and C = {a,e,x}

Here we are given that

A = {a, b, c, d, e}

B = {x, y, z}

C = {a, e, x}

Here we have to verify the eqⁿ given below

n(AUBUC) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C) + n(A∩B∩C)

Here we have to find the cardinality of all the sets and their intersection. So basing on the data given we can find the values for the eqⁿ .

We are given that,

A = {a, b, c, d, e}

B = {x, y, z}

C = {a, e, x}

(A Ո B) = {a, b, c, d, e} Ո {x, y, z} = {Φ}

∵ it’s a Disjoint set and it has no element

(B Ո C) = {x, y, z} Ո {a, e, x} = {x}

(A Ո C) = {a, b, c, d, e} Ո {a, e, x} = {a, e}

(A Ո B Ո C) = {a, b, c, d, e} Ո {x, y, z} Ո {a, e, x} = {Φ}

∵ it’s a Disjoint set as A Ո B has no element

(A ⋃ B ⋃ C) = {a, b, c, d, e} ⋃ {x, y, z} ⋃ {a, e, x}

= {a, b, c, d, e, x, y, z}

So from the above expression we can find the cardinality of all the values,

n(A) = 5

n(B) = 3

n(C) = 3

n(AՈB) = 0

n(BՈC) = 1

n(AՈC) = 2

n(A Ո B Ո C) = 0

n(A ⋃ B ⋃ C) = 8

now using the formula and putting values obtained,

n(AUBUC) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C) + n(A∩B∩C)

⇒ 8 = 5 + 3 + 3 – 0 – 1 – 2 + 0

⇒ 8 = 8 hence verified