2. Let R and S be two relations on the set X. Define R S to be the relation This

Question

2. Let R and S be two relations on the set X. Define R S to be the relation This bowtie operator can be used to define a relational algebra that is a basic tool in the study of databases. (a) Suppose that R and S are relations and R is reflexive. Prove that S R S. (b) Let X be a set as above. Show that there exists a relation I on X with the nice property that for any relation Ron X, we have I R = R I = R. Give an exact description of the relation I. (c) Prove that for any three relations A, B, and C (all over the same set X). we have

Explanation / Answer

(a)

Given R and S be two relations on set X.

The relation R is reflexive means that

The relation S is defined as

The Natural Join between two relations is represented as .

The Natural Join combines all the relations in the two relation sets R and S.

The relations present in the set S are also present in RS.

So, the relation S becomes a proper subset of RS. And R and RS is not equal.

Thus, S RS is true.

(b)

So, I is a identity relation which is reflexive.

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