Question

let X={a,b,c,d} and R={(a,a),(b,b),(a,c)}. write down the minimum number of ordered pairs to be included to R to make it. (1) reflexive. (2) symmetric. (3) transitive. (4) equivalence

Answer 1

Answer:

1. reflexive

 Let X be any non empty set and R be a relaion to X. Now, R is said to be reflexive a is related to a for all a belongs to X.

(ie) Any element that is related to itself is reflexive.

So, in the given Relation, (a,a) and (b,b) are reflexive as they are related to itself. Now if you add (c,c) and (d,d) to the given relation then the whole relation becomes reflexive relation.

2. Symmetric

 In symmeyric relation, a is related to b implies that b is also related to a.

In the given Relation, (a,c) ie: a is related to c, so, if c is also related to a then the relation becomes symmetric.

Therefore you need to add (c,a) to make the relation symmetric.

3. Transitive

In a transitive relation, a is related to b and b is related to c implies that a is also related to c.

In the given relation we have (a,c) but you need to add (a,b) and (b,c) inorder to make it transitive.

4. Equivalence.

If the given relation is said to be reflexive, symmetric and transitive, then the relation is said to be an equivalence relation.

So, when you add (c,c), (d,d), (c,a), (a,b) and (b,c), the relation becomes reflexive, symmetric, and transitive. Therefore, when you add the said relations, the given relation becomes equivalence relation.

Step-by-step explanation:

Answer 2

Answer:

X = {a, b, c, d} R = {(a, a), (b, b), (a, c)}

(i) To make R reflexive we need to include (c, c) and (d, d)

ii) To make R symmetric we need to include (c, a)

(iii) R is transitive

(iv) To make R reflexive we need to include (c, c)

To make R symmetric we need to include (c, c) and (c, a) for transitive

∴ The relation now becomes R = {(a, a), (b, b), (a, c), (c, c), (c, a)}

∴ R is equivalence relation