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Question
Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.
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Solution
(i) Reflexive:
Let a ∈ R, a ≤ a3, which is false.
∴ (a, a) ∉ R
Thus, R is not reflexive.
(ii) Symmetric:
Let a, b ∈ R, and (a, b) ∈ R
⇒ a ≤ b3
Does not imply b ≤ a3
∴ (b, a) ∉ R
Thus, R is not symmetric.
(iii) Transitive:
Let a, b, c ∈ R, consider (a, b) ∈ R and (b, c) ∈ R
⇒ a ≤ b3 and b ≤ c3
⇒ a ≤ c3 is false.
⇒ (a, c) ∉ R
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
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