Section 1.1 & 1.2: Types of Relations

Determine whether each of the following relations are reflexive, symmetric and transitive:

1. Relation \(R\) in the set \(A = \{1, 2, 3, \ldots, 13, 14\}\) defined as
   \[R = \{(x, y) : 3x - y = 0\}\]

2. Relation \(R\) in the set \(\mathbb{N}\) of natural numbers defined as
   \[R = \{(x, y) : y = x + 5 \text{ and } x < 4\}\]

3. Relation \(R\) in the set \(A = \{1, 2, 3, 4, 5, 6\}\) as
   \[R = \{(x, y) : y \text{ is divisible by } x\}\]

4. Relation \(R\) in the set \(\mathbb{Z}\) of all integers defined as
   \[R = \{(x, y) : x - y \text{ is an integer}\}\]

5. Relation \(R\) in the set \(A\) of human beings in a town at a particular time given by
   

1. \(R = \{(x, y) : x \text{ and } y \text{ work at the same place}\}\)
   
2. \(R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}\)
   
3. \(R = \{(x, y) : x \text{ is exactly 7 cm taller than } y\}\)
   
4. \(R = \{(x, y) : x \text{ is wife of } y\}\)
   
5. \(R = \{(x, y) : x \text{ is father of } y\}\)