Let N be the set of natural numbers and a relation R on N be defined by \[R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0\} \]. Then the relation R is :
Solutions · 1
Official solution
x3−3x2y−xy2+3y3=0
\[ \Rightarrow x({x^2} - {y^2}) - 3y({x^2} - {y^2}) = 0\]
\[ \Rightarrow (x - 3y)(x - y)(x + y) = 0\]
Now, x = y \[\forall(x, y) \[\in\]N \[\times\] N so reflexive but not symmetric \& transitive.
See, (3, 1) satisfies but (1, 3) does not. Also (3, 1) \& (1, -1) satisfies but (3, -1) does not.