Let R be the relation on Z given by: for all x, y ∈ Z, xRy if and only if x + y is odd. Determine whether R is reflexive, symmetric, and/or transitive.

Answer 1

xRx. x+x = 2x which isn't odd so not reflexive

xRy then x+y is odd, so y+x is odd, so yRx so symmetric

xRy and yRz. x+y is odd , y+z is odd. x+2y+z is even. So x+z must be even. So not transitive