RELATIONS - TYPES OF RELATIONS( PART 2) - CLASS 12 MATHEMATICS

                                                         Types of Relations

Dear friends, in the previous blog we learnt about the meaning of "RELATION" and meaning of         "REFLEXIVE RELATION". If you have missed click below

Relations - post 1

Now we'll continue with the topic 😊😊

Symmetric  relations:

I wish this relation were symmetric.

( poor zoo zoo).

Let us define a relation in the set of human beings to understand the meaning of symmetric relations

 

 

e.g. A is related to B if A and B live in the same locality, where A and B are human beings.

So, Mr. Mehra is related to Mr. Sharma as both live in the same locality.

So Mr. Mehra can tell Mr. Sharma ,” I am related to you.”

Can Mr. Sharma say the same words to Mr. Mehra?

Or in other words, is Mr. Sharma related to Mr. Mehra with the same relation?

The answer is YES.

Because if “ A “ lives in the neighborhood of “B” then definitely “B” also lives in the neighborhood of “A”

Friends, Such relations are called symmetric relations.

So, we can generalize it like this:

If a is related to b with a fixed relation and b is also related to a with same relation, then we’ll say the relation is symmetric.

( Now you know why poor zoo zoo’s relation should be symmetric)

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Let us discuss one more example

A is related to B if A is father of B, where A and B are human beings.

Now think on this..

   If Mr. Raj is father of  Nikhil , can we say that Nikhil is related to Mr. Raj with the same relation?

The answer is NO.

Nikhil is related to Mr. Raj with the relation “ is son of “

It means Nikhil does not have the same relation with Mr. Raj.

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Ok now let us go to the world of numbers.

So,

R = { (a,b) : 2a + b = 0 } in the set of Real numbers.

Read it like this, a is related to b such that 2a + b = 0

Here,  1 is related to -2

As, 2 (1) + ( -2) = 0 is true 

( see the relation above i.e. a is related to b if 2a + b=0)

Can we say -2 is also related to 1?

Let us see : 2 ( -2) + 1 = -4 + 1 = -3 .

So it is not equal to 0.

Therefore – 2 is no related to 1

 

So, the conclusion is that 1 is related to -2 but -2 is not related to 1

Therefore, it is a relation which is not symmetric.

Generalizing the definition:

So, in simple words if element “a” is related to element “b” and then element “b” is also related to “a” with the same relation, then the relation is called symmetric relation.

 

Transitive  relations:

So, we are at the last type of relations now. “ Transitive relations”.

Now consider an relation in the set of human beings denoted by H.

R = { (a,b) : a is brother of b , a,b  }

Read as a is related to b if a is brother of b , where a and b “ belong to “ set H

                                                                               ( a and b are Human beings)

So, Here Sonu (a) is related to Monu (b) since Sonu is Monu’s brother

Similarly, Monu (b) is related to Gonu (c) since Monu is Gonu’s brother

In place of names we can always call them “a” , “b” and “c”

Now “a” is related to “b” and “b” is related to ”c”.

Now, a question for you

Is “a” related to “c” with the same relation

( or in other words )

( Is Sonu, Gonu’s brother ?)

The answer is “Yes”.

 

I guess even you answered correctly.

So, such relations in which if “a” is related to “b” and “b” is related to “c”, it implies that “a” is related to “c” then relation is called transitive relation.

Okay, one question , is the following relation also transitive?

In the set of all the lines in a 2 dimensional plane

A line “a” is related to “b” if line ”a” is perpendicular  to line “b”

 

Answer in the comments:

So, friends now we have understood three types of relations. Now we can define “EQUIVALENCE RELATIONS” .

A relation which is “reflexive, “symmetric” and “transitive” is termed as an equivalence relation.

So by this last definition we come to the end of this topic. 

FOR NUMERICALS CLICK ON THE LINK

RELATIONS IMPORTANT QUSTIONS

See you later. Stay blessed.😊🙋