**Associative property of sets :**

Here we are going to see the associative property used in sets.

For any two two sets, the following statements are true.

(i) Set union is associative

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

(i) Set intersection is associative

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

Let us look into some example problems based on above properties.

**Example 1 :**

Given, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}, show that

(i) A U (B U C) = (A U B) U C. (ii) Verify (i) using Venn diagram.

**Solution :**

Now, B U C = {3, 4, 5, 6} U {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}

A U (B U C) = {1, 2, 3, 4, 5} U { 3, 4, 5, 6, 7, 8}

= {1, 2, 3, 4, 5, 6, 7, 8} ----- (1)

Now, (A U B) = {1, 2, 3, 4, 5} U {3, 4, 5, 6} = {1,2,3,4,5,6}

(A U B) U C = {1,2,3,4,5,6} U {5,6,7,8}

= {1, 2, 3, 4, 5, 6, 7, 8} ----- (2)

(1) = (2)

**Example 2 :**

Let A = {a,b,c,d}, B = {a,c,e} and C = {a,e}.

(i) Show that A n (B n C) = (A n B) n C. (ii) Verify (i) using Venn diagram

**Solution :**

Now, B n C = {a,c,e} n {a, e} = {a, e}

A n (B n C) = {a, b, c ,d} U {a, e}

= { a } ----- (1)

(A n B) = {a,b,c,d} n {a,c,e}

= {a, c}

(A n B) n C = {a, c} n {a,e}

= {a} ----- (2)

(1) = (2)

**Example 3 :**

For A = {x ; x is a prime factor of 42}, B = {x | 5 x ≤ 12, x ∊ N} and C = {1, 4, 5, 6}, verify A U (B U C) = (A U B) U C

**Solution :**

**A = {2, 3, 7}, B = {5, 6, 7, 8, 9, 10, 11, 12} and C = {1, 4, 5, 6}**

**(B U C) = ****{5, 6, 7, 8, 9, 10, 11, 12} U ****{1, 4, 5, 6}**

** = {1, 2, 4, 6, 8, 9, 10, 11, 12}**

**A U (B U C) = ****{2, 3, 7} U ****{1, 2, 4, 6, 8, 9, 10, 11, 12}**

** = {1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12} ----(1)**

**(A U B) = ****{2, 3, 7} U {5, 6, 7, 8, 9, 10, 11, 12}**

**= {2, 3, 5, 6, 7, 8, 9, 10, 11, 12}**

**(A U B) U C = ****{2, 3, 5, 6, 7, 8, 9, 10, 11, 12} U ****{1, 4, 5, 6}**

** = {1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12} ----(2)**

**(1) = (2)**

- Venn diagram A U B
- Venn diagram A n B
- Venn diagram of A'
- Venn diagram of B'
- Venn diagram of (AUB)'
- Venn diagram of (AnB)'
- Venn diagram of A\B
- Venn diagram of B\A

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