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Question
Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric ?
Solution
Let R be the set such that R = {(a, b) : a, b ∈ R; a ≥ b}
Reflexivity :
Let a be an arbitrary element of R.
⇒ a∈ R
⇒ a = a
⇒ a ≥ a is true for a = a
⇒ (a, a) ∈ R
Hence, R is reflexive.
Symmetry :
Let (a, b) ∈ R
⇒ a ≥b is same as b ≤ a, but not b ≥ a
Thus, (b, a) ∉ R
Hence, R is not symmetric .
Transitivity :
Let (a, b) and (b, c) ∈ R
⇒a ≥ b and b ≥ c
⇒ a ≥ b ≥ c
⇒ a ≥ c
⇒ ( a, c) ∈ R
Hence, R is transitive .
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