# Sets and Functions - Reflexive - Symmetric - Antisymmetric - Transitive

by Shine
(Saudi Arabia)

Let R be the relation on the set of real numbers defined by x R y iff x-y is a rational number . Determine whether R is reflexive , symmetric , antisymmetric and /or transitive

### Comments for Sets and Functions - Reflexive - Symmetric - Antisymmetric - Transitive

 May 07, 2012 Sets and Functions - Reflexive - Symmetric - Antisymmetric - Transitive by: Staff Question: by Shine (Saudi Arabia) Let R be the relation on the set of real numbers defined by x R y iff x-y is a rational number. Determine whether R is reflexive, symmetric, antisymmetric and /or transitive Answer: Definitions: Reflexive: relation R is REFLEXIVE if xRx for all values of x Symmetric: relation R is SYMMETRIC if xRy implies yRx Antisymmetric: relation R is ANTISYMMETRIC if xRy and yRx implies x = y Transitive: relation R is TRANSITIVE if xRy and yRz implies xRz ----------------------------------------------------- x R y iff x - y is a rational number Reflexive? YES. x-x = 0 is rational, so xRx for all x. Symmetric? YES. xRy, x-y is rational. y - x = -(x - y) is also rational (the negative of a rational number is also a rational number). Therefore, yRx is also true. R is symmetric. Antisymmetric? NO. 1 - 0 = 1 and 0 - 1 = -1 are rational, but 1 ≠ 0. So 1R0 and 0R1 not antisymmetric. Transitive? YES. If xRy and yRz, then x - y and y - z are rational. The sum of two rational numbers is also a rational number. Therefore, x - z = (x - y) + (y - z) is rational, and xRz is true. R is transitive. Thanks for writing. Staff www.solving-math-problems.com

 Apr 16, 2020 Reflexive - Symmetric - Antisymmetric - Transitive NEW by: Anonymous For integers x and y , x S y if and only if x+5 y is divisible by 6.