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# OPERATIONS ON SETS

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(1) Union of Sets: The union of two sets A and B is the set of all the elements which are either in A or in B or in both A and B. It is denoted by $$A\cup B$$.

Let $$\text{A}=\left\{ 1,\,\,2,\,\,\,3 \right\}$$

$$\text{B}=\left\{ 2,\,\,\,4,\,\,6 \right\}$$

$$\displaystyle \therefore \text{A}\cup \text{B}=\left\{ 1,\,\,2,\,\,3,\,\,4,\,\,6 \right\}$$

(2) Intersection of Sets: The intersection of two sets A and B is the set of all the elements which are common to both A and B. It is denoted by $$\text{A}\cap \text{B}$$

Let $$\text{A}=\left\{ 1,\,\,2,\,\,3 \right\}$$

$$\text{B}=\left\{ 2,\,\,4,\,\,6 \right\}$$

$$\therefore \text{A}\cap \text{B}=\left\{ 2 \right\}$$

(3) Difference of Two Sets: $$\left( \text{A}\,-\,\text{B} \right)$$ is the set of all those elements of A which are not elements of B.

$$\text{A}\,-\,\text{B}=\left\{ x:x\in \text{A}\,\text{and}\,x\,\notin \,\text{B} \right\}$$

Similarly, $$\left( \text{B}\,-\,\text{A} \right)$$ is the set of all those elements of B which are not elements of A.

$$\text{B}-\text{A}=\left\{ x:x\in \text{B}\,\text{and}\,x\in \text{A} \right\}$$

Let $$\text{A}=\,\left\{ 1,\,\,3,\,\,5,\,\,6,\,\,7 \right\}$$ and $$B=\left\{ 2,\,\,3,\,\,4,\,\,5 \right\}$$

Then, $$\text{A}-\text{B}=\,\left\{ 1,\,\,6,\,\,7 \right\}$$

$$\displaystyle \text{B}-\text{A}\,\text{=}\,\left\{ 2,\,\,4 \right\}\,$$

(4) Symmetric Difference of Two Sets: Let A and B be two sets. $$\left( \text{A}-\text{B} \right)\cup \left( \text{B}-\text{A} \right)$$ is the symmetric difference between two sets. It is denoted by $$\text{A}\,\Delta \,\text{B}$$ .

Consider $$\text{A}=\left\{ 1,2,3,4,5,6 \right\}$$ and $$\text{B}=\left\{ 1,3,5,7,9 \right\}$$

Then, $$\text{A}-\text{B}=\left\{ 2,4,6 \right\}$$

$$\text{B}-\text{A}=\left\{ 7,9 \right\}$$

$$\therefore \,\,\,\,\text{A}\,\Delta \,\text{B}=\left( \text{A}-\text{B} \right)\cup \left( \text{B}-\text{A} \right)=\left\{ 2,4,6,7,9 \right\}$$

(5) Complement of a set: The complement of a set A is the set of all those elements of the universal set U which are not element of A. It is denoted by $$\text{A }\!\!’\!\!\text{ }$$ or $${{\text{A}}^{\text{c}}}$$ or $$\displaystyle \left( \text{U}-\text{A} \right)$$.

Let $$\text{A}=\left\{ 2,4,6,8 \right\}$$

$$\text{U=}\left\{ 1,2,3,4,5,6,7,8 \right\}$$

Then, $$\text{A }\!\!’\!\!\text{ }=U-A\text{=}\left\{ 1,2,3,5,7 \right\}$$

RESULTS ON COMPLEMENT

1. $$\text{U}’=\phi$$
2. $$\phi \text{ }\!\!’\!\!\text{ =}\,U$$
3. $$\left( \text{A }\!\!’\!\!\text{ } \right)’=\text{A}$$
4. $$\text{A}\cup \text{A }\!\!’\!\!\text{ }=\,U$$
5. $$\text{A}\cap \text{A }\!\!’\!\!\text{ }=\,\phi$$

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