Question

Let A={1,2,3}. then find the number of relations containing (1,2)and (1,3) which are reflexive and symmetric but not transitive.​

Answer 1

Step-by-step explanation:

for reflexive

if a belongs to R

then (a, a) must belongs to R

to make given relation reflexie

(1,1),(2,2),(3,3) belongs to R

for symmetric

if (a, b) belongs to R then (b, a) must belongs to R

imply (2,1),(3,1) belongs to R

R = {1,1 2,2 3,3 2,1 3,1 1,2 1,3 }

smallest set of relation which satisfy given relation

now it is reflexive for all element 1,2,3

toake it completely symmeteic we add

(1,2),(1,3)(2,3)

(1,2),(1,3) already in relation only one pair left (2,3)

but if we add this also then our relation satify condition of transitive relation

now for transitive all pairs are here only one pair left (2,3) which make it transitive.

Answer => 1

Answer 2

Step-by-step explanation:

We Have :-

A = { 1 , 2 , 3 }

To Find :-

Then find the number of relations containing ( 1 , 2 ) and ( 1 , 3 ) which are reflexive and symmetric but not transitive

Solution :-

The Possible Pairs that can be formed are :-

{ ( 1 , 1 ) ; ( 1 , 2 ) ; ( 1 , 3 ) ; ( 2 , 1 ) ; ( 2 , 2 ) ; ( 2 , 3 ) ; ( 3 , 1 ) ; ( 3 , 2 ) ; ( 3 , 3 ) }

In Symmetric :-

If ( a , b ) is in relation then ( b , a ) should also be in relation .

so ,

( 1 , 2 ) ; ( 2 , 1 ) both should be in relation

( 1 , 3 ) ; ( 3 , 1 ) both should be in relation

In Reflexive :-

( a , a ) should be in relation

( 1 , 1 ) ; ( 2 , 2 ) ; ( 3 , 3 ) should be in relation

In Transitive :-

( a , b ) and ( b , c ) are in relation then ( a , c ) should be in relation

So relation without transitive is

R = { ( 1 , 2 ) ; ( 1 , 3 ) ; ( 1 , 1 ) ; ( 2 , 2 ) ; ( 3 , 3 ) ; ( 2 , 1 ) ; ( 3 , 1 ) }