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Solution for Three Relations R1, R2 And R3 Are Defined on a Set A = {A, B, C} as Follows: R1 = Find Whether Or Not Each of the Relations R1, R2, R3, R4 On A Is (I) Reflexive (Ii) Symmetric and (Iii) Transitive. - CBSE (Science) Class 12 - Mathematics

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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, R2R3R4 on is (i) reflexive (ii) symmetric and (iii) transitive.

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 symmetric.

Transitive:
Here,

(a, b)R1, (b, c)R1 and also (a, c)R1

So, R1 is transitive.

(ii) R2

Reflexive: Clearly (a,aR2 . So, R2 is reflexive.

Symmetric: Clearly (a,a⇒ (a,aR. 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, cR3but (c,bR3

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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Solution Three Relations R1, R2 And R3 Are Defined on a Set A = {A, B, C} as Follows: R1 = Find Whether Or Not Each of the Relations R1, R2, R3, R4 On A Is (I) Reflexive (Ii) Symmetric and (Iii) Transitive. Concept: Types of Relations.
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