Question

In: Computer Science

select the relation that is an equivalence relation. THe domain set is (1,2,3,4). a. (1,4)(4,1),(2,2)(3,3) b....

select the relation that is an equivalence relation. THe domain set is (1,2,3,4).

a. (1,4)(4,1),(2,2)(3,3)

b. (1,4) (4,1)(1,3)(3,1)(2,2)

c. (1,4)(4,1)(1,1)(2,2)(3,3)(4,4)

d. (1,4)(4,1)(1,3)(3,1)(1,1)(2,2)(3,3)(4.4)

Expert Solution

Before directly going to the answer let us check what is an equivalence relation.

Equivalence relation set is nothing but a relation which is

Symmetry, transitive and reflexive.

Symmetry =aRb->bRa for all a, b belongs to A(domain set).
Transitive =aRb and bRc - > aRc for a, b, c belongs to A(domain set).
Reflexive = aRa for all a belongs A(domain set) .

Now let us take each of the given relations in the question.

a) for reflexive test every element of the domain set should have a (a, a) relation where a belongs to domain set. But here (2,2)&(3,3) belongs to relation.. But the relations does not cover every element, ie. (1,1)&(4,4)

So the relations is not reflexive.

Since this is not reflexive there is no chance for the relation to be a equivalence relation. So we don't want to check other criteria..

a is not a equivalence relation.

b) a mentioned above let us check for reflexive.

And this is clear thatvthe relation don't have a (a, a) relation. So b is not reflexive and there by not equivalence relation.

c) here (1,1),(2,2),(3,3)& (4,4) belongs to the relations. So the relation is reflexive.

Then let us check for Symmetry relation. As mentioned above for Symmetry aRb->bRa for a, b belongs to A(domain set).

Here (1,4)&(4,1)belongs to relation set R and satisfy the conditions. So this is an symmetric relation.

Finally let us check for transitive . As mentioned above to be a transitive relation aRb and bRc - > aRc for all a, b, c belongs to A(domain set).

Here (1,4),(4,1) & (1,1) belongs to the relation. And hence this is also a transitive relation.

Hence c is symmetric, reflexive and transitive this is an equivalence relation.

d) this is clear that it is an reflexive relation. ((1,1),(2,2),(3,3),(4,4)).

This is clear that it is also a Symmetrical relation((1,4),(4,1) also (1,3),(3,1)).

Finally for transitive it is clear that the relation also a transitive reaction. ((1,4),(4,1),(1,1)).

So Hence d is symmetric, reflexive and transitive this is an equivalence relation.

venereology answered 2 hours ago

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