GATE Overflow Test Series | Mock GATE | Test 2 | Question: 15

Question

Let $R$ be a relation on set $A.$ Assuming $A$ has at least three elements and $R^{-1}$ denotes the inverse relation, state which of the following statement is/are TRUE.
Statement $1:$ If $R$ is reflexive then $R \cap R^{-1}$ is not empty.
Statement $2:$ If $R$ is symmetric then $R = R^{-1}$

• A. Both statements are true
• B. Statement $1$ is true and Statement $2$ is false
• C. Both statements are false
• D. Statement $2$ is true and Statement $1$ is false

Answer 1

Lets assume set $A = \{a,b,c\}$

A reflexive relation $R$ on $A$ will have the ordered pairs $(a,a), (b,b)$ and $(c,c).$ And these pairs will always be there in $R^{-1}$ as well. So, $R \cap R^{-1}$ is always non empty. Statement $1$ is TRUE.

Now, is $R$ is symmetric, for every pair $(a,b)$ the inverse relation pair $(b,a)$ is also in $R.$ That is $R = R^{-1}.$ So, statement $2$ is TRUE.

So, the correct answer is $(A).$

Comments

IF R={(1,2)(2,1)}

R^-1={(2,1)(1,2)}

R intersection R^-1 is not  empty but it is not reflexive .

answer should be only B.

@Deepak Poonia and @Arjun Suresh 7 @Arjun Suresh 6@Arjun Suresh 2 sir please check this once

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@JayRathi Option A says If R is reflexive Then prove that stmst 

Acc to your ex, R itself is not reflexive

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ok got it bro thank you

Answer 2

Consider the set $A = \left \{ 1,2,3,4,..... ,n \right \}$

if $R$ is a relation defined on $A$ then $R^{-1}$ is the relation which is obtained by flipping the tuples in $R$

example: Let $R = \left \{ (1,2),(3,4),(3,1) \right \}$ then $R^{-1} = \left \{ (2,1),(4,3),(1,3) \right \}$

 

(i) Statement 1: If $R$ is reflexive then $R \cap R^{-1}$ is not empty

Proof : If $R$ is reflexive then $(x,x) \in R$  $\forall _{x} \in A$

$R^{-1}$ will then also contain all the reflexive pairs

Hence $R \cap R^{-1}$ is guaranteed to contain all reflexive pairs hence it will be reflexive, so this is TRUE

 

(ii) Statement 2: If $R$ is symmetric then $R = R^{-1}$ is not empty

Proof : If $R$ is symmetric then $(x,y),(y,x) \in R$ 

$R^{-1}$ will then contain $(y,x),(x,y)$ $\equiv$ $R^{-1}$ will then contain $(x,y),(y,x)$

$\therefore R = R^{-1}$, hence this statement is also TRUE

 

More properties of inverse relations : https://proofwiki.org/wiki/Inverse_Relation_Properties