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Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which of the following is TRUE?

Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which of the following is TRUE? Correct Answer R is neither symmetric nor anti-symmetric

The correct answer is option 2.

Concept:

A binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation.

Symmetric relation:

In symmetric relation, if a=b is true then b=a is also true. In other words, a relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. 

Anti-symmetric relation:

The relation R is said to be antisymmetric on a set A if xRy and yRx hold when x = y. Or it can be defined as, relation R is antisymmetric if either (x,y)∉R or (y,x)∉R whenever x ≠ y.

Explanation: 

The binary relation  R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}

Symmetric relation: 

The R is not symmetric relation because for (x, y) element there is no element (y, x) present and for (z, y) element there is no element (y, z) present.

Hence it is not symmetric relation.

Anti-symmetric relation:

The R is not anti-symmetric relation because for (x, z) element there is an element (z, x) is present. 

Hence it is not Anti-symmetric relation.

Hence the correct answer is R is neither symmetric nor anti-symmetric.

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