Which of the following relations is symmetric but neither reflexive nor transitive for a set A= {a, b, c}?

Concept:

Reflexive: A relation is said to be reflexive if (a, a) ∈ R, for every a ∈ A.

Ex. The relation R in the set {1, 2, 3} given by 

R = {(1,1), (2,2), (3,3)} is reflexive.

Symmetric: A relation is said to be symmetric, if (a, b) ∈ R, then (b, a) ∈ R.

Ex. The relation R in the set {a, b, c} given by

R = {(a,b), (b,a), (b,c), (c,b)} is symmetric.

Transitive: A relation is said to be transitive if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

Ex. The relation R in the set {1, 2, 3} given by

R = {(1,2), (2,3), (1,3)}

Explanation:

Given set A = {a, b, c}

For option (1):

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

For the given set A, R is symmetric as aRb ⇒ bRa and  bRc ⇒ cRb

Since every element is not related to itself (a, a). (b, b) and (c, c) are not the elements of R so the relation R is not reflexive.

Also, (a, b) and (b, c) ∈ R. but (a ,c) is not the element of R. 

So, the relation R is not transitive so option 1 is correct.

For option (2):

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

For the given set A, R is not symmetric.

Since (a, b) ∈ R but (b, a) ∉ R.

For option (3):

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

For the given set A, R is reflexive.

Since every element is related to itself.

For option (4):

​R= {(a, b), (a, c), (a. d)}

For the given set A, R is not symmetric.

Since (a, b) ∈ R but (b, a) ∉ R.

∴ The correct option is (1).