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 - SThen -A- R 2 is symmetric but it is not transitive-B- both R 1 and R 2 are transitive-C- both R 1 and R 2 are not symmetric-D- R 1 is not symmetric but it is transitive-

As (b, c) ∈ R_1 but (c, b) ∉R_ 1 ∴ R_1 is not symmetric. As (b, c), (c, a)∈ R_1 but (b, a) ∉R_1 nbsp; ∴ R_1 is not transitive. We can see that R_2 is symmetr ...

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Standard XII

Mathematics

Transitive Relations

Consider the ...

Question

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 :

both

R_{1}

and

R_{2}

are not symmetric.

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R_{1}

is not symmetric but it is transitive.

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R_{2}

is symmetric but it is not transitive.

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both

R_{1}

and

R_{2}

are transitive.

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Solution

The correct option is C $R_{2}$ is symmetric but it is not transitive.
As $(b, c) \in R_{1}$ but $(c, b) \notin R_{1}$
$∴ R_{1}$ is not symmetric.
As $(b, c), (c, a) \in R_{1}$ but $(b, a) \notin R_{1}$
$∴ R_{1}$ is not transitive.

We can see that $R_{2}$ is symmetric.
Also, $(c, a), (a, b) \in R_{2}$ but $(c, b) \notin R_{2}$
$∴ R_{2}$ is not transitive .

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Similar questions

Q. 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.

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. 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. Let

R_{1}

and

R_{2}

be two relations defined as follows:

R_{1} = {(a, b) \in R^{2} : a^{2} + b^{2} \in Q}

and

R_{2} = {(a, b) \in R^{2} : a^{2} + b^{2} \notin Q} .

Then

Q. Let

R_{1}

and

R_{2}

be two relations defined on a non empty set A. Which of the following statements is false? Give reasons in support of your answer.
(a) If

R_{1}

and

R_{2}

are reflexive, then so is

R_{1} \cap R_{2}

(b) If

R_{1}

and

R_{2}

are symmetric, then so is

R_{1} \cup R_{2}

R_{1}

and

R_{2}

are transitive, then so is

R_{1} \cup R_{2}

(d) If

R_{1}

and

R_{2}

are transitive, then so is

R_{1} \cap R_{2}

(e) If

R_{1}

and

R_{2}

are symmetric, then so is

R_{1} \cap R_{2}

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Consider the following two binary relations on the set A={a,b,c}: R1={(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)} and R2={(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c)}. Then :

The correct option is C R2 is symmetric but it is not transitive.As (b,c)∈R1 but (c,b)∉R1 ∴R1 is not symmetric. As (b,c),(c,a)∈R1 but (b,a)∉R1 ∴R1 is not transitive. We can see that R2 is symmetric. Also, (c,a),(a,b)∈R2 but (c,b)∉R2 ∴R2 is not transitive .

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 :