First slide

Relations XII

Question

Let $R_{1}$ be a relation defined by $R_1 = {(a, b) | a \geq b, a, b \in R} .$ Then $R_{1}$ is

Easy

A

An equivalence relation on R
B

Reflexive, transitive but not symmetric
C

Symmetric, Transitive but not reflexive
D

Neither transitive not reflexive by symmetric

Solution

For any $a \in R,$ we have $a \geq a,$ therefore the relation R is reflexive but it is not symmetric as $(2, 1) \in R$ but
$(1, 2) \notin R .$ The relation R is transitive also, because $(a, b) \in R, (b, c) \in R$ imply that $a \geq b and b \geq c$ which is turn imply that $a \geq c .$

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