Solution to Determine whether the following relation is reflexive, symmetric, antisymmetric and/or transitive. (x, y) ∈ R … - Sikademy
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Archangel Macsika

Determine whether the following relation is reflexive, symmetric, antisymmetric and/or transitive. (x, y) ∈ R if x ≥ y, where R is the set of positive integers.

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Let us determine whether the relation (x, y) ∈ R  if x ≥ y is reflexive, symmetric, antisymmetric and/or transitive.

Since x\ge x for each positive integer x, we conclude that the relation is reflexive.

Taking into account that (2,1)\in R but (1,2)\notin R, we conclude that the relation is not symmetric.

If (x,y)\in R and (y,x)\in R, then x\ge y and y\ge x. It follows that x\ge y\ge x, and hence  y=x.We conclude that the relation is antisymmetric.

Since (x,y)\in R and (y,z)\in R imply x\ge y and y\ge z, we conclude that x\ge y\ge z, and thus  x\ge z. It follows that (x,z)\in R, and thus the relation is transitive.

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