Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation `R` in the set `A = {1, 2, 3, ..., 13, 14}` defined as `R = {(x, y) : 3x – y = 0}`
(ii) Relation `R` in the set `N` of natural numbers defined as `R = {(x, y) : y = x + 5` and `x lt 4}`
(iii) Relation `R` in the set `A = {1, 2, 3, 4, 5, 6}` as `R = {(x, y) : y` is divisible by `x`}
(iv) Relation `R` in the set `Z` of all integers defined as `R = {(x, y) : x – y` is an integer}
(v) Relation `R` in the set `A` of human beings in a town at a particular time given by
(a) `R = {(x, y) : x` and `y` work at the same place}
(b) `R = {(x, y) : x` and `y` live in the same locality}
(c) `R = {(x, y) : x` is exactly `7` cm taller than `y`}
(d) `R = {(x, y) : x` is wife of `y`}
(e) `R = {(x, y) : x` is father of `y`}

Explanation

To determine whether each of the given relations is reflexive, symmetric, and transitive, we will analyze each relation step by step.

### (i) Relation R in the set A = {1, 2, 3, ..., 14} defined as R = {(x, y) : 3x - y = 0}

1. **Reflexive**: A relation is reflexive if (a, a) is in R for every a in A. For (x, x) to be in R, we need 3x - x = 0, which simplifies to 2x = 0. This only holds true for x = 0, which is not in the set A. Therefore, R is **not reflexive**.

2. **Symmetric**: A relation is symmetric if whenever (a, b) is in R, then (b, a) is also in R. If (x, y) is in R, then 3x - y = 0 implies y = 3x. For (y, x) to be in R, we need 3y - x = 0, which implies x = 3y. This is not generally true. For example, (1, 3) is in R, but (3, 1) is not. Therefore, R is **not symmetric**.

...