Q. 1B 4.1( 63 Votes )

Relation R in the set N of natural numbers defined as
R = {(x, y) : y = x + 5 and x < 4}

Determine if the given relation is reflexive, symmetric and transitive.

Answer :

It is given that Relation R in the set N of natural numbers defined as

R = {(x, y) : y = x + 5 and x < 4}

Clearly, 
R = {(1, 6), (2, 7), (3, 8)}


REFLEXIVE
A relation is said to be reflexive if (x, x)  ϵ  R, where x is from domain. 
we can see that (1,1) ∉ R

⇒ R is not reflexive.


SYMMETRIC 
A relation is said to be symmetric if (y, x)  ϵ  R whenever (x, y) ϵ R.

Now, (1,6) ϵ r but (6,1) ∉ R.

⇒ R is not symmetric.


TRANSITIVE

Now, since there is no pair in R such that (x, y) and (y, z) ∈R, then (x, z) cannot belong to R.

R is not transitive.


Therefore, R is neither reflexive, nor symmetric, nor transitive.

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