Test by Bikram | Mock GATE | Test 1 | Question: 29

Question

Let $R$ be a binary relation on the set of all positive integers such that $R = \{ (a, b) \mid a - b \text{ is an odd positive integer} \}$

$R$ is :

• A. an anti-symmetric relation    
• B. a reflexive and symmetric relation     
• C. an equivalence relation  
• D. a partial ordering relation

Answer 1

R is equivalence relation If It Is reflexive, symmetric and transitive.
R is partial ordering reiation If It is refiexive, antisymmetrlc and transitive.
as a - b is positive integer, b - a is not positive hence anti-symmetric  .
a - b is positive (odd), b - c is positive (odd) but (a - c) is even hence not transitive. So R is anti-symmetric

Comments

but anti-symmetric allows diagonal elements, and in this case, (xRx) is not positive odd(it is 0).?

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it allows but doesnt necessarily need them, thus it can be one of the condition for equivalence and PO, and is not asymmetric