Let a relation `R` on the set `N` of natural numbers be defined at `(x ,y) in R ,` if and only if `x^2-4x y+3y^2=0` for all `x ,y in Ndot` The relation `R` is a. Reflexive b. Symmetric c. Transitive d. Equivalence Relation

Answer

2

Explanation

We have `R={(x, y) : x^(2)-4xy+3y^(2)=0, x, y in N}`
Let `x in N, x^(2)-4.x.x+3x^(2)=0`
`:. (x, x) in R :. R` is reflexive
we have `(3)^(2)-4(3)(1)+3(1)^(2)=9-12+3=0`
or `(3, 1) in R`
Also `1^(2)-4(1)(3)+3(3)^(2)=1-12+27 ne 0`
`:. (1, 3) in R, :. R` is not symmetric
again `(9, 3) in R` because
`9^(2)-4(9)(3)+3(3)^(2)=108-108=0`
and `(3, 1) in R` because
`(3)^(2)-4(3)(1)+3(1)^(2)=12-12=0`
and `(9, 1) in R` if `9^(2)-4(9)(1)+3(1)^(2)=0`
if `84-36=0` which is not possible
`:. (9, 3), (3, 1) in R` and `(9, 1) in R :. R` is not transitive
`:.` Relation R is reflexive but neither symmetric nor transitive.