\(n\left( A\cup B\cup C \right) \)= n(A) + n(B) + n(C) - \(n\left( A\cap B \right) -\left( B\cap C \right) -n\left( A\cap C \right) +\left( A\cap B\cap B \right) \)

For any three sets A, B and C
(i) \(A\cap \left( B\cap C \right) =\left( A\cap B \right) \cap C\)
(ii) \(A\cup \left( B\cup C \right) =\left( A\cup B \right) \cup C\)

Given A = {0, 2, 4,6, 8} , B = {2, 3, 5, 7} and C = {5, 6, 7, 8}
First, we find \(B\cap C\) = {5, 7} ,\(A\cup (B\cap C)\) = {0, 2, 4, 5, 6, 7, 8} .......... (1)
Next, \(A\cup B\) = {0,2,3,4,5,6,7, 8},  \(A\cup C\) = {0, 2, 4, 5, 6, 7, 8}
Then, \((A\cup B)\cap (A\cup C)\) = {0, 2, 4, 5, 6, 7, 8} .............(2)
From (1) and (2), it is verified that \((A\cup \left( B\cap C \right) =(A\cup B)\cap (A\cup C)\).

A = {x : x = 2ⁿ, n ∈ W, n < 4}
x = 2⁰ = 1
x = 2¹ = 2
x = 2² = 4
x = 2³ = 8
ஃ A = {1,2,4,8}
B = {x : x = 2n, n∈N and n ≤ 4}
x = 2 \(\times\) 1 = 2
x = 2 \(\times\) 2 = 4
x = 2 \(\times\) 3 = 6
x = 2 \(\times\)4 = 8
ஃ B = {2,4,6,8}
C = {0,1,2,5,6}
Associative property of intersection of sets
A∩(B∩C) = (A∩B)∩C
B∩C = {2,6}
A∩(B∩C) = {1,2,4,6,8}∩{2,6}
= {2} .............(1)
A∩B = {1,2,4,8} ∩ {2,4,6,8} = {2,4,8}
(A∩B) ∩C = {2,4,8} ∩ {0,1,2,5,6}
= {2} ............(2)
From (1) and (2), It is verified that A∩(B∩C) = (A∩B)∩C

Associative Property of intersection of sets
A⋂(B⋂C) = (A⋂B)⋂C
B⋂C = {√3, √5, 6, 13} ∩ {√2, √3, √5, 9} = {√3,√5}
A⋂( B⋂C) = {-11,√2,√5,7} ⋂ {√3,√5} = {√5}...(1)
A⋂B = {-11,√2,√5,7}√{√3,√5,6,13} = {√5}
(A∩B)∩C = {√5} ∩ {√2,√3,√5,9} = {√5} ...(2)
From (1) and (2), it is verified that A∩(B∩C) = (A∩B)∩C

Associative Property of union of sets
AU(BUC) = (AUB)UC
BUC = {m, n, q, s, t} U {m, n, p, q, s} = {m, n, p, q, s, t}
AU(BUC) = {p, q, r, s} U {m, n, p, q, s, t} = {m, n, p, q, r, s, t} ...(1)
(AUB) = {p, q, r, s} U {m, n, q, s, t} = {p, q, r, s, m, n, t}
(AUB)UC = {p, q, r, s, m, n, t} U {m, n, p, q, s} = {p, q, r, s, m, n, t}...(2)
From (1) & (2)
It is verified that AU(BUC) =( AUB)UC

Commutative Property of union of sets
(A∪B) = (B∪A)
Here P = {3,4,5,6}, Q = {√3,√5,√6}
P∪Q = {3,4,5,6}∪{√3,√5,√6} = {3,4,5,6,√3,√5,√6} ...(1)
Q∪P = {√3,√5,√6}∪{3,4,5,6} = {√3,√5,√6,3,4,5,6} ...(2)
(1)=(2)
ஃ P∪Q = Q∪P
ஃ It is verified that union of sets is commutative.
Commutative Property of intersection of sets
(P⋂Q) = (Q∩P)
P∩Q = {3,4,5,6}∩{√3,√5,√6} = { } ...(1)
Q∩P = {√3,√5,√6}⋂{3,4,5,6} = { } ...(2)
From (1) and (2)
P∩Q = Q∩P
ஃ It is verified that intersection of sets is commutative.

(i) (P∪Q)∪R
(P∪Q) = {1, 2, 5, 7, 9} U {2, 3, 5, 9, 11} = {1,2,3,5,7,9,11}
(P∪Q)∪R = {1,2,3,5,7,9,11} U {3,4,5,7,9} = {1,2,3,4,5,7,9,11}
(ii) (P⋂Q)⋂S
(P⋂Q) = {1,2,5,7,9} ⋂ {2,3,5,9,11} = {2,5,9}
(P⋂Q)⋂S = {2,5,9} ⋂ {2,3,4,5,8} = {2,5}
(iii) (Q⋂S)⋂R
(Q⋂S) = {2,3,5,9,11} ⋂ {2,3,4,5,8} = {2,3,5}
(Q⋂S)⋂R = {2,3,5} ⋂ {3,4,5,7,9} = {3,5}

Now,  \((B\cap C)=\left\{ \frac { 1 }{ 4 } ,2,\frac { 5 }{ 2 } \right\} \)
\(A\cap (B\cap C)=\left\{ \frac { 1 }{ 4 } ,2 \right\} \)    .......(1)
Then, \(A\cap B=\left\{ 0,\frac { 1 }{ 4 } ,\frac { 3 }{ 4 } ,2 \right\} \)
\((A\cap B)\cap C=\left\{ \frac { 1 }{ 4 } ,2 \right\} \)        .......(2)
From (1) and (2), it is verified that
\((A\cap B)\cap C=A\cap (B\cap C)\).

Given, A = {2, 3, 4, 5},B = {2, 3, 5,7} and C = {1, 3, 5}
Now, \(B\cup C\) = {1,2, 3,5,7}
\(A\cup (B\cup C)\) ={1, 2, 3, 4, 5, 7}        .....(1)
Then, \(A\cup B\) = {2, 3, 4,5,7}
\((A\cup B)\cup C\) = {1,2, 3, 4, 5, 7}    ...(2)
From (1) and (2), it is verified that \(A\cup (B\cup C)\) = \((A\cup B)\cup C\).