Sets and Functions - Reflexive - Symmetric - Antisymmetric - Transitive
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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

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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


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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
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