What are the corresponding relation for the following three examples? a. reflexi

Question

What are the corresponding relation for the following three examples? a. reflexive and symmetric but not transitive, b. reflexive and transitive but not symmetric, c. symmetric and transitive but not reflexive.

Answer format: a

Example I. Consider the relation that makes no two elements related.

Example II. Consider the set of vertices in some non-complete, non-empty graph where we make two vertices related if they are adjacent or the same vertex.

Example III. Consider the vertices in a digraph where we have aRb if there is some possibly empty path from a to b. For example, suppose we have only the set {0, 1} and the relations 0R0, 0R1, and 1R1.

Explanation / Answer

Relation for three with example:

A relation R in a set A is subset of A × A. Thus empty set and A × A are two extreme relations.

(i) A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = A × A.

(ii) A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A.

(iii) A relation R in A is said to be reflexive if aRa for all aA, R is symmetric if aRb bRa, a, b A and it is said to be transitive if aRb and bRc aRc a, b, c A. Any relation which is reflexive, symmetric and transitive is called an equivalence relation.

Example 1

Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?

Solution :

R is reflexive and symmetric, but not transitive since for (1, 0) R and (0, 3) R whereas (1, 3) R.

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