Q. 1B4.1( 200 Votes )

# Relation R in the set N of natural numbers defined asR = {(x, y) : y = x + 5 and x < 4}Determine if the given relation is reflexive, symmetric and transitive.

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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