GATE Overflow | Mathematics | Test 1 | Question: 1

Question

A relation $R$ is defined as $xRy$ , if $x$ and $y$ are NOT equal. This relation $R$ is

• A. symmetric but not reflexive
• B. symmetric and transitive but not reflexive
• C. an equivalent relation
• D. none of reflexive or symmetric or transitive

Comments

I believe option is B , since it is transitive as well .
Why is the answer A ?

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xRy and yRx then xRx so it won't be transitive

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Why option a is correct pls explain

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@kavita_joshi

xRy and yRx then xRx so it is not  transitive .

read this slide -->  http://www3.cs.stonybrook.edu/~pfodor/courses/CSE215/L14-Relations.pdf

Answer 1

relation hold iff x!=y

1 :for reflexive: xRx  (x!=x) false ;

2: for symmetric xRy=yRx ( x!=y and y!=x )  true

3: for transitive lets : (x,y)=(1,2) (y,z) =(21) (x,z)=(11)

here xRy ,yRz hold but xRx doesn't (as 1=1 )

Comments

So , a fact can be derived that every transitive relation is reflexive also.

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@satendra That is incorrect.

$A=\left \{1,2,3 \right \}$

Relation R on A = $A=\left \{(1,2) ,(2,1), (1,1) \right \}$

Transitive but not reflexive.

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No it is not transitive because we have (2,1) and (1,2).

Therefore we need (2,2) as well to make it Transitive.

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No every Transitive relation is not always reflexive. It depends on how we have defined our relation.

eg let A = {1,2,3}

Let R be a relation from A to A, such that R = {(1,2), (2,1), (1,1), (2,2)}

Now this relation is Symmetric and Transitive but not Reflexive.

Answer 2

• Can't be reflexive.
  
   
• Symmetric, yes. Because if xRy then yRx.
  
   
• Transitive? Let's see. if xRy and yRz then xRz. True.
  
  if xRy and yRx then xRx... So, not transitive as xRx doesn't belong to the "not equal to" relation.

Option A