UGC NET Computer Science December 2019 | Question 55

Question 55

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Let P be the set of all people. Let R be a binary relation on P such that (a,b) is in R if a is a brother of b. Is R symmetric, transitive, an equivalence relation, a partial order relation?

1. 1. NO,NO,NO,NO
2. 2. NO,NO,YES,NO
3. 3. NO,YES,NO,NO
4. 4. NO,YES,YES,NO

Explanation

Below is the list of the required definations:

• To prove that R is symmetric, if (a, b) ∈ R → (b, a) ∈ R
• To prove transitivity, if (a, b),(b, c) ∈ R then (a, c) ∈ R
• R is a equivalence relation if R is reflexive, symmetric and transitive.
• A binary relation R is a partial order if and only if the relation is reflexive, antisymmetric and transitive.
• Relation R is Reflexive if (a, a) ∈ R for every a ∈ A

R is not symmetric as a is a brother of b. So b should be brother of a which is not necessary. b might be sister of a.

R is transitive as a is brother of b and b is brother of c. We can say that a is brother of c.

R is not reflexive as a is brother of a(himself).

As it's proved that reflexive property is not satisfied for given relation R. Reflexive property needs to be satified for relation R to be equivalent as well as Partial order. This implies that R is not equivalence as well as Partial order.

So, we can conclude as below:
Is R symmetric? - NO
Is R transitive? - YES
Is R equivalence relation? - NO
Is R partial order relation? - NO

Hence, Option 3 (NO,YES,NO,NO) is the right answer.

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