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Question
Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.
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Solution
(i) R1
Reflexive:
Clearly, (a, a), (b, b) and (c, c) ∈ R1
So, R1 is reflexive.
Symmetric:
We see that the ordered pairs obtained by interchanging the components of R1 are also in R1.
So, R1 is not symmetric.
Transitive:
Here,
(a, b)∈R1, (b, c)∈R1 and also (a, c)∈R1
So, R1 is transitive.
(ii) R2
Reflexive: Clearly (a,a) ∈R2 . So, R2 is reflexive.
Symmetric: Clearly (a,a) ∈R ⇒ (a,a) ∈R. So, R2 is symmetric.
Transitive: R2 is clearly a transitive relation, since there is only one element in it.
iii) R3
Reflexive:
Here,
(b, b)∉ R3 neither (c, c) ∉ R3
So, R3 is not reflexive.
Symmetric:
Here,
(b, c) ∈R3, but (c,b) ∉R3
So,R3isnotsymmetric.
Transitive:
Here, R3 has only two elements. Hence, R3 is transitive.
(iv) R4
Reflexive:
Here,
(a, a) ∉ R4, (b, b)∉ R4 (c, c)∉ R4
So, R4 is not reflexive.
Symmetric:
Here,
(a, b)∈ R4, but (b,a) ∉ R4.
So, R4 is not symmetric
Transitive:
Here,
(a, b)∈R4, (b, c)∈R4, but (a, c)∉R4
So, R4 is not transitive.
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