Question

In: Advanced Math

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising...

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation. (c) (x1,y1)R(x2,y2) in R×R if x1∗y2 = x2∗y1.

Solutions

Expert Solution

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Reflexive:

Since hence

Thus is reflexive.

Symmetric:

Assume that

then

Now

Thus R is symmetric.

Transitive:

Assume that

-----------------------(1)

Again assume that

------------------------(2)

Now multiplying (1) by and (2) by we have

Thus is transitive.

Combining we find that is an equivalence relation.

Equivalence Relation:

Notice that

Our set of equivalence classes will be those sets of ordered pairs of real numbers who when divided equal one another. For example, if we consider the equivalence class ,then Included in this would also be and so on.

Similarly if we consider equivalence class , then Included in this would also be and so on.

Colby Messinger answered 1 year ago

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