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- cR2-a- aR3-b- cR4-a- b-b- c-c- aFind whether or not each of the relations R1- R2- R3- R4 on A is i reflexive ii symmetric and iii transitive-

(i) R1 Reflexive: Clearly, (a, a), (b, b) and (c, c) #8712;R1 So, R1 is reflexive. Symmetric: We see that the ordered pairs obtained by interchanging the com ...

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Standard XII

Mathematics

Equivalence Relation

Three relatio...

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) $\in$ 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,
$(a, b) \in R_{1}, (b, c) \in R_{1} and also (a, c) \in R_{1}$
So, R₁ is transitive.

(ii) R₂
Reflexive: Clearly $(a, a) \in R_{2}$ . So, R₂ is reflexive.
Symmetric: Clearly $(a, a) \in R \Rightarrow (a, a) \in R$ . So, R₂ is symmetric.
Transitive: R₂ is clearly a transitive relation, since there is only one element in it.

(iii) R₃
Reflexive:
Here,
$(b, b) \notin R_{3} n e i t h e r (c, c) \notin R_{3}$
So, R₃is not reflexive.

Symmetric:
Here,
$(b, c) \in R_{3}, but (c, b) \notin R_{3} So, R_{3} is not symmetric .$

Transitive:
Here, R₃ has only two elements. Hence, R₃ is transitive.

(iv) R₄
Reflexive:
Here,
$(a, a) \notin R_{4}, (b, b) \notin R_{4} (c, c) \notin R_{4} So, R_{4} is not reflexive .$

Symmetric:
Here,
$(a, b) \in R_{4}, b u t (b, a) \notin R_{4} . So, R_{4} is not symmetric .$

Transitive:
Here,
$(a, b) \in R_{4}, (b, c) \in R_{4}, b u t (a, c) \notin R_{4} So, R_{4} is not transitive .$

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Similar questions

Q. Three relations

R_{1}, R_{2}

, and

R_{3}

are defined on set A ={ a, b, c} as follows:
(i)

R_{1}

= { (a, a), (a, b), (a, c), (b, c), (c, a), (b, b), (c, b), (c, c)}
(ii)

R_{2}

= {( a, b), (b, a), (a, c), (c, a)}
(iii)

R_{3}

= {(a, b), (b, c), (c, a)}
Discuss each of them from the point of view of being reflexive, symmetric and transitive.

Q. Let

A = {1, 2, 3}

and let

R_{1} = {(1, 1), (1, 3) (3, 1) (2, 2), (2, 1) (3, 3)}

R_{2} = {(2, 2), (3, 1), (1, 3)}

R_{3} = {(1, 3), (3, 3)}

R_{4} = A \times A .

Find which of the given each of the relations

R_{1}, R_{2}, R_{3}, R_{4},

on A satisfy all the given properties:

(a) reflexive (b) symmetric (c) transitive

Q. Test whether the following relations R₁, R₂, and R₃ are (i) reflexive (ii) symmetric and (iii) transitive:
(i) R₁ on Q₀ defined by (a, b) ∈ R₁ ⇔ a = 1/b.
(ii) R₂ on Z defined by (a, b) ∈ R₂ ⇔ |a – b| ≤ 5
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Q. Consider the following two binary relations on the set

A = {a, b, c} :

R_{1} = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R_{2} = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)} .

Then :

Q. Is the relation

R

defined on a set

A = {a, b, c}

as follows:

R = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}

is (i) reflexive (ii) symmetric (iii) transitive

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