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Question
Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.
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Solution
(i) Reflexive:
R = {(a, b) : a ≤ b2}
Let a ∈ R
a ≤ a2 which is false.
(a, a) ∉ R
∴ R is not reflexive.
(ii) Symmetric:
Let a, b ∈ R and (a, b) ∈ R
a ≤ b2 and b ≤ a2, which is false.
(a, b) ∈ R, but (b, a) ∉ R
∴ R is not symmetric.
(iii) Transitive:
Let a, b, c ∈ R
Consider (a, b) ∈ R, (b, c) ∈ R
a ≤ b2 and b ≤ c2
a ≤ c2 is false.
(a, c) ∉ R
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
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