Question

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

Answer 1

Answer:

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 $a \in X, (a,a) \in R$.

R is said to be symmetric if for every $(a, b) \in R, (b, a) \in R$.

R is said to be transitive if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

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

Reflexive: $(a, a), (b, b), (c, c) \in R$

Therefore, R is reflexive.

Symmetric: $(a, b) \in R \implies (b, a) \in R$

Therefore R is symmetric.

Transitive: $(a, b) \in R \ and \ (b, c) \in R$ 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: $(a, a), (b, b) \ and \ (c, c) \in R$

Therefore, R is reflexive.

Symmetric: $(a, b) \in R \ but \ (b, a) \not \in R$

Therefore R is not symmetric.

Transitive: $(a, a), (a, b) \in R$ and $(a, b) \in R$.

Therefore, R is transitive.

Therefore, R is reflexive and transitive but not symmetric .

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

Reflexive: $(a, a) \in R$ 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: $(a, b) \in R$ and $(b, a) \in R$

Therefore R is symmetric.

Transitive: $(a, a), (a, b) \in R$ and $(a, b) \in R$.

Therefore, R is transitive.

Therefore, R is symmetric and transitive but not reflexive .