**Formulas used in set operations :**

Here we are going to see formulas used in set operations.

For any two finite sets A and B, we have the following useful results

(i) n(A) = n(A - B) + n(A n B)

(ii) n(B) = n(B - A) + n(A n B)

(iii) n(A U B) = n(A - B) + n(A n B) + n(B - A)

(iv) n(A U B) = n(A) + n(B) - n(A n B)

(v) n(A U B) = n(A) + n(B), when A n B = null set

(vi) n(A) + n(A') = n(U)

Let us look into some example problems based on the above concept.

**Example 1 :**

If n(A n B) = 5, n(A U B) = 35, n (A) = 13, find n (B) .

**Solution :**

**Using the formula,**

**n(A U B) = n(A) + n(B) - n(A n B)**

**35 = 13 + n (B) - 5**

** 35 = 8 + n (B) **

**Subtract 8 on both sides**

**35 - 8 = 8 + n (B) - 8**

**27 = n (B) **

**Hence the value of n (B) is 27.**

**Example 2 :**

If n (A) = 26, n (B) = 10, n (A U B) = 30, n (A') = 17, find n (A n B) and n (U) .

**Solution :**

**Using the formula,**

**n(A U B) = n(A) + n(B) - n(A + B)**

**30 = 26 + 10 ****- n(A n B)**

**30 = 36 ****- n(A n B)**

**Subtract 36 on both sides**

**30 - 36 = ****36 ****- n(A n B) - 36**

**-6 = ****- n(A n B)**

**Hence the value of n ****(A n B)**** is 6**

**n (U) = n (A) + n (A')**

** = 26 + 17 ==> n ****(U) = 43**

**Hence the value of n (U) is 43.**

**Example 3 :**

If n(U) = 38, n(A) = 16, n(A n B) = 12, n(B') = 20

find n(A U B).

**Solution :**

n(A U B) = ** n(A) + n(B) - n(A n B)**

**In order to use the above formula to find n (AUB), we need the value of n (B).**

**n (B) + n (B') = n (U)**

**n (B) + 20 = 38**

**n (B) = 38 - 20 ==> 18**

**n(A U B) = 16 + 18 - 12**

** = 34 - 12**

** = 22**

**Hence the value of ****n(A U B) is 22.**

**Example 4 :**

A and B are two sets such that n(A - B) = 32 + x, n(B - A) = 5x and n(A n B) = x Illustrate the information by means of a Venn diagram. Given that n(A) = n(B) . Calculate (i) the value of x (ii) n(A U B) .

Solution :

Formula to find n (A) is n(A) = n(A - B) + n(A n B)

Formula to find n (B) is n(B) = n(B - A) + n(A n B)

n(A) = 32 + x + x ==> 32 + 2x ---(1)

n(B) = 5x + x ==> 6x -----(2)

Given that n (A) = n (B)

32 + 2x = 6x

Subtract 2x on both sides

32 + 2x - 2x = 6x - 2x

32 = 4x

Divide by 4 on both sides, we get

32/4 = 4x /4

x = 8

(i) Hence the value of x is 8.

(ii) n(A U B) = n(A - B) + n(A n B) + n(B - A)

= 32 + x + x + 5x

= 32 + 7x

Applying the value of x,

= 32 + 7(8)

= 32 + 56

= 88

- 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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