GATE CSE 1995 | Question: 1.19

Question

Let $R$ be a symmetric and transitive relation on a set $A$. Then

• A. $R$ is reflexive and hence an equivalence relation
• B. $R$ is reflexive and hence a partial order
• C. $R$ is reflexive and hence not an equivalence relation
• D. None of the above

Comments

Observation from that Question

If aRb, then since R is symmetric:

bRa

and since R is transitive:

aRb and bRa  ⟹  aRa

Similarly,

bRb

So any element that is related to something becomes reflexive on itself. But this still does not mean every element of A is reflexive.

That is why reflexivity does not automatically follow.

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Try this variation : https://gateoverflow.in/1522/gate-cse-1999-question-3

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Let R be a symmetric and transitive relation on a set A. Then
if it mentioned that for every element x E A ; X R Y i.e x relates to some element Y then R will become reflexive

Answer 1

The answer is $D$.

Let $A=\{1,2,3\}$ and relation $R=\{(1,2),(2,1),(1,1),(2,2)\}. R$ is symmetric and transitive but not reflexive. Because $(3,3)$ is not there.

Comments

@sujeetkumar, bro please read option C carefully, it saying that R is reflexive which is not true in above question.

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Under rare circumstances i see comments with more upvotes than the original answer, mainly due to intuitive approach. Thanks !

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for Non empty Base set.

because ,

Empty Relation on Empty set is Reflexive,symmetric,transitive,Ireflexive,Asymmetric.

Answer 2

Answer: D

Let A = {(1,2),(2,1),(1,1)}

A is symmetric and transitive but not reflexive as (2,2) is not there.

Comments

According to the example you have assumed, thought the answer remains correct, but you must include (2,2) into the relation as well because of transitivity. and may be you can change the set A to , A={1,2,3}

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simply, take (1,1) as relation  which is symmetric and transitive, but for reflexive it should have other pair ( b,b) &(c,c) if i consider set {1,2,3}

hence option D

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The explanation is not right!

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@rajshree you must include (2,2) in your explanation

Answer 3

here ans should be D

explanation:

here the relation is symmetric and transitive. if relation is symmetric and transitive then it need not necessariy be reflexive;i.e. it may or may not be reflexive. therefore ans is D

Answer 4

We can take an empty set { } which is both symmetric and and transitive but not reflexive because diagonal elememts are not present in the set so not reflexive.

Answer 5

The relation $R=\phi$ on a non-empty set is symmetric, transitive but not reflexive.

$1.(a,a)\notin R$ because R is empty set | Reflexive $\times$

 

$2.(a,b)\in R$ is always FALSE (as R is the empty set), the conditional statement $((a,b)\in R)\rightarrow B$ is true for any statement B. Hence Symmetric.

 

$3.(a,b)\in R$ is always FALSE (as R is the empty set) and since $(b,c)\in R$ is always FALSE (as R is the empty set), the statement $(a,b)\in R$ $\wedge$ $(b,c)\in R$ is also always false.

Then the conditional statement $( (a,b)\in R \wedge (b,c)\in R)\rightarrow B $ is true for any statement B. Hence Transitive

Source : Kenneth H. Rosen

Correct Answer : D 

Comments

Condition for transitivity is correct?

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@gatecse

sir, is it Okay ?

Answer 6

The correct option is D None of the above

A relation which is symmetric and transitive, need not be reflexive relation.
 

(i) R={}: on the set A={a,b}. The relation R is symmetric and transitive but not reflexive.
 

(ii) R={(a,a).(b,b)}; on the set A={a,b}
The relation R is symmetric, transitive and also reflexive.
∴ A relation is transitive and symmetric relation but need not be reflexive relation.