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Question
Test whether the following relation R1 is (i) reflexive (ii) symmetric and (iii) transitive :
R1 on Q0 defined by (a, b) ∈ R1 ⇔ a = 1/b.
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Solution
1. Reflexivity:
Let a be an arbitrary element of R1. Then,
a ∈ R1
⇒ a ≠1/a for all a ∈ Q0
So, R1 is not reflexive.
2. Symmetry:
Let (a, b) ∈ R1 Then,
(a, b) ∈ R1
a =`1/b`
⇒ `b = 1/a`
⇒ `(b, a) ∈ R_1`
So, R1 is symmetric.
3. Transitivity:
Here,
(a, b) ∈ R1 and (b, c) ∈R2
⇒ `a = 1/b and b = 1/c `
⇒ `a = 1/(1/c)=c`
⇒ `a ≠ 1/c`
⇒ (a ,c) ∉ R1
So, R1 is not transitive.
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