Amongst the properties [reflexivity, symmetry, anti-symmetry, transitivity] the relation R = {(x, y) in N x N | x * y <= x^2} satisfies _________

GATE 1994 · Discrete Mathematics · Relations · medium

Answer: R satisfies reflexivity, anti-symmetry, and transitivity (R is a partial order on N). It does NOT satisfy symmetry.

1. Simplify the condition: Dividing both sides of x*y <= x^2 by x (positive), we get y <= x. So R = {(x, y) | y <= x} on N (excluding 0 if N starts at 1).
2. Check reflexivity: x <= x is always true, so (x, x) in R for all x. R is reflexive.
3. Check symmetry and antisymmetry: Take x = 3, y = 1: y <= x so (3,1) in R. But x <= y? 3 <= 1 is false, so (1, 3) not in R. Symmetry fails. For antisymmetry: if y <= x and x <= y then x = y — antisymmetry holds.
4. Check transitivity: If z <= y and y <= x, then z <= x by transitivity of <=. So (x, z) in R. R is transitive.