**Distributive property of set :**

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

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

(i) Union distributes over intersection

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

(ii) Intersection distributes over union

A n (B u C) = (A n B) U (A n C)

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

**Example 1 :**

Let A = {0, 1, 2, 3, 4}, B = {1, - 2, 3, 4, 5, 6} and C = {2, 4, 6, 7}.

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

**Solution :**

(B n C) = {1, - 2, 3, 4, 5, 6} n {2, 4, 6, 7}.

= {4, 6}

A U (B n C) = {0, 1, 2, 3, 4} U {4, 6}

= {0, 1, 2, 3, 4, 6} --------(1)

(A U B) = {0, 1, 2, 3, 4} U {1, - 2, 3, 4, 5, 6}

= {-2, 0, 1, 2, 3, 4, 5, 6}

(A U C) = {0, 1, 2, 3, 4} U {2, 4, 6, 7}

= {0, 1, 2, 3, 4, 6, 7}

(AUB) n (AUC) = {-2, 0, 1, 2, 3, 4, 5, 6} n {0, 1, 2, 3, 4, 6, 7}

= {0, 1, 2, 3, 4, 6} --------(2)

**Example 2 :**

For A = {x : - 3 ≤ x < 4, x ∊ R}, B = {x ; x < 5, x ∊ N} and C = {- 5, - 3, - 1,0,1,3}, Show that A n (B U C) = (A n B) U (A n C)

**Solution :**

A = {x : - 3 ≤ x < 4, x ∊ R} that is, A consists of all real numbers from – 3 upto 4 but 4 is not included.

A = {-3, -2, -1, 0, 1, 2, 3}

B = {x ; x < 5, x ∊ N} = {1, 2, 3, 4}

(B U C) = {1, 2, 3, 4} U {- 5, - 3, - 1, 0, 1, 3}

= {-5, -3, -1, 0, 1, 2, 3, 4}

An(B U C) = {-3, -2, -1, 0, 1, 2, 3} n {-5, -3, -1, 0, 1, 2, 3, 4}

= {-3, -1, 0, 1, 2, 3} -------(1)

(A n B) = {-3, -2, -1, 0, 1, 2, 3} n {1, 2, 3, 4}

= {1, 2, 3}

(A n C) = {-3, -2, -1, 0, 1, 2, 3} n {- 5, - 3, - 1, 0, 1, 3}

= {-3, -1, 0, 1, 2, 3}

(A n B) U (A n C) = {1, 2, 3} U {-3, -1, 0, 1, 2, 3}

= {-3, -1, 0, 1, 2, 3} -------(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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