**Practice Problems on Properties of Set Theory :**

Here we are going to see some based on properties of set theory.

**Commutative property :**

For any two sets A and B

(i) A U B = B U A

(ii) A n B = B n A

**Associative property :**

For any three sets A, B and C

(i) A U (B U C) = (A U B) U C

(ii)A n (B n C) = (A n B) n C

**Question 1 :**

If P = {1,2,5,7,9}, Q = {2, 3,5,9,11}, R = {3, 4,5,7,9} and S = {2, 3, 4,5, 8}, then find (i) (P U Q) U R (ii) (P n Q) n S (iii) (Q n S) n R

**Solution :**

P = {1, 2, 5, 7, 9}

Q = {2, 3, 5, 9, 11}

R = {3, 4, 5, 7, 9}

S = {2, 3, 4, 5, 8}

(i) (P U Q) U R

(P U Q) = {1, 2, 3, 5, 7, 9, 11}

(P U Q) U R = {1, 2, 3, 4, 5, 7, 9, 11} -------(1)

(ii) (P n Q) n S

P n Q = {2, 5, 9}

(P n Q) n S = {2, 5} ------(2)

(iii) (Q n S) n R

Q n S = {2, 3, 5}

(Q n S) n R = {3, 5} ------(3)

**Question 2 :**

Test for the commutative property of union and intersection of the sets

P = { x : x is a real number between 2 and 7} and

Q = { x : x is an irrational numbers between 2 and 7}

**Solution :**

P = {3, 4, 5, 6}

Q = {2.01......, .............., 6.990,..............}

P U Q = Q U P

P n Q = Q n P

**Question 3 :**

If A = {p, q, r, s}, B = {m, n, q, s, t} and C = {m, n, p, q, s}, then verify the associative property of union of sets.

**Solution :**

**Associative property for union :**

A U (B U C) = (AU B) U C

B U C = {m, n, p, q, s, t}

A U (B U C) = {m, n, p, q, r, s, t} ------(1)

(AU B) = {m, n, p, q, r, s, t}

(AU B) U C = {m, n, p, q, r, s, t} ------(2)

(1) = (2)

Hence proved.

**Question 4 :**

Verify the associative property of intersection of sets for A = {−11, √2, √5, 7}, B = {√3, √5, 6, 13} and C = {√2, √3, √5, 9}.

**Solution :**

**Associative property for intersection :**

A n (B n C) = (A n B) n C

B n C = {√3, √5}

A n (B n C) = { √5 } ------(1)

(A n B) = { √5 }

(A n B) n C = { √5 } ------(2)

(1) = (2)

Hence proved.

**Question 5 :**

If A = {x : x = 2^{n}, n ∈ W and n < 4}, B = {x : x = 2n, n ∈ N and n ≤ 4} and C = {0, 1, 2, 5, 6}, then verify the associative property of intersection of sets.

**Solution :**

**A = {x : x = 2 ^{n}, n ∈ W and n < 4}**

**A = {1, 4, 8}**

**B = {x : x = 2n, n ∈ N and n ≤ 4}**

**B = { 2, 4, 6, 8 }**

**C = {0, 1, 2, 5, 6}**

**Associative property for intersection :**

A n (B n C) = (A n B) n C

(B n C) = {2, 6}

A n (B n C) = { } ---(1)

(A n B) = {4, 8}

(A n B) n C = { } ---(2)

Hence proved.

After having gone through the stuff given above, we hope that the students would have understood, "Practice Problems on Properties of Set Theory".

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