(ii) Relation R in the set N of natural numbers defined asR = {(x, y) : y = x + 5 and x < 4}. Determine whether each of the following relations are reflexive, symmetric andtransitive without taking example

To determine whether the relation R is reflexive, symmetric, and transitive, we need to understand the properties of these types of relations:

1. Reflexive: A relation R in a set A is said to be reflexive if every element is related to itself. That is, (a, a) ∈ R for every a ∈ A. In this case, for the relation R to be reflexive, we would need every (x, x) pair such that y = x + 5 and x < 4. However, there is no natural number x that satisfies y = x + 5 for y = x. Therefore, the relation R is not reflexive.

2. Symmetric: A relation R in a set A is said to be symmetric if for every (a, b) ∈ R, we have (b, a) ∈ R. In this case, for the relation R to be symmetric, we would need every (x, y) pair to also have a corresponding (y, x) pair such that y = x + 5 and x < 4. However, if we take an example pair (x, y) = (1, 6) from R, the pair (6, 1) does not satisfy y = x + 5 for x < 4. Therefore, the relation R is not symmetric.

3. Transitive: A relation R in a set A is said to be transitive if for every (a, b) ∈ R and (b, c) ∈ R, we have (a, c) ∈ R. In this case, for the relation R to be transitive, we would need every pair (x, y) and (y, z) to also have a corresponding (x, z) pair such that y = x + 5 and x < 4. However, given the constraints of the relation, it is impossible to find such pairs in R. Therefore, the relation R is not transitive.

In conclusion, the relation R = {(x, y) : y = x + 5 and x < 4} in the set N of natural numbers is neither reflexive, symmetric, nor transitive.