Let R be a relation from Q to Q defined by `R={(a,b):a,b in Q and a,b in Z}. Show that `{a,a) in R for all a in Q ,`{a,b} in R implies` that `{b,a} in R, {a,b} in R and {b,c} in R implies that {a,c} in R` `

Explanation

To solve the problem, we need to show three properties of the relation \( R \) defined on the set of rational numbers \( Q \) such that \( R = \{(a, b) : a, b \in Q \text{ and } a - b \in Z\} \), where \( Z \) is the set of integers.

### Step 1: Show that \( (a, a) \in R \) for all \( a \in Q \)

To show this, we need to check if the pair \( (a, a) \) satisfies the condition of the relation \( R \).

- Let \( a \in Q \).
- Calculate \( a - a = 0 \).
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