The given relation is defined on the set of real numbers. `a R b iff |a| = |b|`
. Find whether these relations are reflexive, symmetric or transitive.

Explanation

To determine whether the relation defined by \( a R b \) if and only if \( |a| = |b| \) is reflexive, symmetric, or transitive, we will analyze each property step by step.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( a \) in the set, \( a R a \) holds true.

- For our relation, we need to check if \( |a| = |a| \).
- This is always true since the absolute value of any number is equal to itself.

**Conclusion:** The relation is reflexive.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( a R b \) holds, then \( b R a \) also holds.

- We start with the assumption that \( a R b \) which means \( |a| = |b| \).
- From this equality, we can conclude that \( |b| = |a| \), which means \( b R a \) holds true.

**Conclusion:** The relation is symmetric.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( a R b \) and \( b R c \) hold, then \( a R c \) must also hold.

- Assume \( a R b \) and \( b R c \). This means:
- \( |a| = |b| \) (1)
- \( |b| = |c| \) (2)
- From (1) and (2), we can deduce that \( |a| = |c| \).

**Conclusion:** The relation is transitive.

### Final Conclusion
Since the relation is reflexive, symmetric, and transitive, we conclude that the relation \( R \) is an equivalence relation.

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