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 \(\text{A}=\left\{ 1,\,\,2,\,\,\,3 \right\}\)

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

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

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(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 \(\text{A}=\left\{ 1,\,\,2,\,\,3 \right\}\)

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

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

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(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, \(\text{A}-\text{B}=\,\left\{ 1,\,\,6,\,\,7 \right\}\)

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

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(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, \(\text{A}-\text{B}=\left\{ 2,4,6 \right\}\)

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

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

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(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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