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 show the properties of the relation \( R \) defined on the rational numbers \( \mathbb{Q} \), we need to prove three statements based on the definition of \( R \). The relation \( R \) consists of all ordered pairs \( (a, b) \) such that \( a, b \in \mathbb{Q} \) and \( a - b \in \mathbb{Z} \) (the integers).

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

1. Let \( a \) be any rational number.
2. Consider the pair \( (a, a) \).
3. Calculate \( a - a = 0 \).
4. Since \( 0 \) is an integer (i.e., \( 0 \in \mathbb{Z} \)), we have \( a - a \in \mathbb{Z} \).
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