Show that the relation \(R\) in \(R\) defined as \(R=\{(a, b): a \leq b\}\), is reflexive and transitive but not symmetric.
Expert Answer
\(A=(-\infty, \infty)\) or \(R\)
\(R=\{(a, b): a \leq b\}\)
(a) Reflexive : \(R=\{(a, a): a \leq a\}\) so reflexive.
(b) Symmetric : \(R=\left\{\left(a_1, a_2\right): a_1 \leq a_2\right\}\)
\(R=\left(a_2, a_1\right): a_2 \leq a_1\) so not symmetric.
(c) Transitive : \(R=\left\{\left(a_1, a_2\right): a_1 \leq a_2\right\}\) and
\(R=\left\{\left(a_2, a_3\right): a_2 \leq a_3\right\}\)
Thus, \(a_1 \leq a_2 \leq a_3\)
\(\Rightarrow a_1 \leq a_3\) so transitive.
Verified Answer
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