For each part, give a relation that satisfies the condition. a. reflexive and symmetric but not transitive b. reflexive and transitive but not symmetric c. symmetric and transitive but not reflexive

For the set X = {a, b, c}, the following three relations satisfy the required conditions in (a), (b) and (c) respectively.

(a)R = {(a,a), (b,b), (c, c), (a, b), (b, a), (b, c), (c, b)} is reflexive and symmetric but not transitive .

(b)R = {(a, a), (b, b), (c, c), (a, b)} is reflexive and transitive but not symmetric .

(c)R = {(a,a), (a, b), (b, a)} is symmetric and transitive but not reflexive .

Step-by-step explanation:

Before, we go on to check these relations for the desired properties, let us define what it means for a relation to be reflexive, symmetric or transitive.

Given a relation R on a set X,

R is said to be reflexive if for every .

R is said to be symmetric if for every .

R is said to be transitive if and , then .

(a) Let R = {(a,a), (b,b), (c, c), (a, b), (b, a), (b, c), (c, b)}.

Reflexive:

Therefore, R is reflexive.

Symmetric:

Therefore R is symmetric.

Transitive: but but (a,c) is not in  R.

Therefore, R is not transitive.

Therefore, R is reflexive and symmetric but not transitive .

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

Reflexive:

Therefore, R is reflexive.

Symmetric:

Therefore R is not symmetric.

Transitive: and .

Therefore, R is transitive.

Therefore, R is reflexive and transitive but not symmetric .

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

Reflexive: but (b, b) and (c, c) are not in R

R must contain all ordered pairs of the form (x, x) for all x in R to be considered reflexive.

Therefore, R is not reflexive.

Symmetric: and

Therefore R is symmetric.

Transitive: and .

Therefore, R is transitive.

Therefore, R is symmetric and transitive but not reflexive .

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step-by-step explanation:

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