Intersection of Sets

Intersection is an operation on set . It is just opposite to union. It is very useful concept in mathematics.It is very important concept in set theory. Before we going to learn about intersection ,, we need to understand some basic concept, like Set: A set is a collection of data. its data is known as its members or elements. We represent the set by capital letters A, B, C, X, Y, Z, etc.  We use the concept of set in daily life. Like a team have five member, so this is a set.

Example:

X = { 2, 3, 8, 9 } and Y = { 5, 12, 9, 16 },  so X and Y are two sets.

Now we are going to understand the concept of  Intersection of set. It has represented by the symbol `nn` .

Example:

If we want to find the intersection of A and B ,  the common part of the sets A and B are the intersection of A and B.it is represented as A `nn` B.That is if a element is present in both A and B then that will be there in the intersection of A and B It will be more clear with the below figure.

Let A and B are two sets. Then the intersection of A and B can be shown as below-

intersection of sets

intersection of sets                         

The intersection of A and B is denoted by A∩B and read as “ A intersection B”.

Thus, A∩B = {x : x ϵ A and x ϵ B}.

Clearly, x ϵ A∩B   i.e.,        x ϵ A and x ϵ B

In the above figure, the shaded area represents A∩B.

In the same way, if A1, A2, ………An is a finite family of sets, then their intersection is represented by A1∩A2∩……∩An.

How to find the intersection of sets

For finding the intersection of two sets, we usually select those elements which are common in both the sets. If there are three sets then we select those elements which are common in all three sets. Hence, if there are n number of sets then we select only those elements which are common in all the n sets. In this way, we find the intersection of sets.

Intersecting set: Two set A and B are said to be intersecting if `A nn B != O/`

Disjoint set: Two set A and B are said to be disjoint if `A nn B = O/`

Problem 1 :

If A={1,3,4,6,9} and B={2,4,6,8} find(A ∩ B) , What do you conclude?

Solution :

We have given that  A = {1,3,4,6,9}

and B = {2,4,6,8}

we have to find the intersection of A ∩ B.

so,   A ∩ B = {1,3,4,6,9} ∩ {2,4,6,8}

A `uu` B = {4,6}  Answer.

Problem 2:

If A={1,3,5,7,9} and B={2,4,6,8} , find(A ∩ B)  What do you conclude ?

Solution: 

 We have

A ∩ B ={ 1,3,5,7,9} ∩ {2,4,6,8}

= Φ

If no data match in both the sets so both the sets known as disjoint sets, and If no data match so we write Φ this Symbol.

Thus ,A and B are disjoint sets .

Example where we deal with more than two sets

Problem 3:

If A = {1, 2, 3, 4, 5, 6, 7}, and B = {2, 4, 6, 8, 10} and C = {4, 6, 7, 8, 9, 10, 11}, then find A∩B and A∩B∩C.

Solution:

 Given sets

A = {1, 2, 3, 4, 5, 6, 7}

B = {2, 4, 6, 8, 10}

C = {4, 6, 7, 8, 9, 10, 11}

First we have to find A∩B, After finding A∩B , we treat as a single set.

For A∩B, we select those elements which are common in sets A and B. So,

A∩B = {2, 4, 6}

For (A∩B)∩C, we select those elements which are common in sets A∩B and  C. So,

(A∩B)∩C = {4, 6}

so, A∩B∩C = {4,6 ) Answer.

Problem 4:

If A={1,3,5,7,9},B={2,4,6,8}and C={2,3,5,7,11},find(A ∩ B) and(A ∩ C) What do you conclude?

Solution:  

We have given that

A = {1,3,5,7,9}

B = {2,4,6,8}

and C = {2,3,5,7,11}

A ∩ B={1,3,5,7,9} ∩ {2,4,6,8}

=Φ Answer

Thus ,A and B are disjoint sets

and A∩C = {1,3,5,7,9} ∩ {2,3,5,7,11}

= {3,5,7} Answer.

Thus ,A and B are disjoint sets while A and C are intersecting sets



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