Which of the following is not an equialence relation on I (set of all integers) ?

Options

A. a R b iff a+b is even
B. a R b iff a-b is even
C. a R b is iff `altb`
D. a R b iff a=b

Answer

To determine which of the given relations is not an equivalence relation on the set of integers \( I \), we need to analyze each relation based on the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step 1: Analyze the first relation \( a + b \) is even.

1. **Reflexivity**: For any integer \( a \), \( a + a = 2a \), which is even. Thus, \( aRa \) holds.
2. **Symmetry**: If \( a + b \) is even, then \( b + a \) is also even (since addition is commutative). Thus, if \( aRb \), then \( bRa \).
3. **Transitivity**: If \( a + b \) is even and \( b + c \) is even, then both can be expressed as \( 2k \) for some integers \( k \). Adding these gives \( (a + b) + (b + c) = a + c + 2b \), which is also even. Thus, if \( aRb \) and \( bRc \), then \( aRc \).

**Conclusion**: This relation is an equivalence relation.

### Step 2: Analyze the second relation \( a - b \) is even.

1. **Reflexivity**: For any integer \( a \), \( a - a = 0 \), which is even. Thus, \( aRa \) holds.
2. **Symmetry**: If \( a - b \) is even, then \( b - a \) is also even (since \( b - a = -(a - b) \)). Thus, if \( aRb \), then \( bRa \).
3. **Transitivity**: If \( a - b \) is even and \( b - c \) is even, then both can be expressed as \( 2k \) for some integers \( k \). Adding these gives \( (a - b) + (b - c) = a - c \), which is also even. Thus, if \( aRb \) and \( bRc \), then \( aRc \).

**Conclusion**: This relation is an equivalence relation.

### Step 3: Analyze the third relation \( a < b \).

1. **Reflexivity**: For any integer \( a \), \( a < a \) is false. Thus, \( aRa \) does not hold.
2. **Symmetry**: If \( a < b \), then \( b < a \) is false. Thus, if \( aRb \), then \( bRa \) does not hold.
3. **Transitivity**: If \( a < b \) and \( b < c \), then \( a < c \) holds. Thus, if \( aRb \) and \( bRc \), then \( aRc \).

**Conclusion**: This relation is not an equivalence relation because it fails reflexivity and symmetry.

### Step 4: Analyze the fourth relation \( a = b \).

1. **Reflexivity**: For any integer \( a \), \( a = a \) holds. Thus, \( aRa \) holds.
2. **Symmetry**: If \( a = b \), then \( b = a \) holds. Thus, if \( aRb \), then \( bRa \).
3. **Transitivity**: If \( a = b \) and \( b = c \), then \( a = c \) holds. Thus, if \( aRb \) and \( bRc \), then \( aRc \).

**Conclusion**: This relation is an equivalence relation.

### Final Answer:
The relation that is not an equivalence relation on the set of integers is the third option: \( a < b \).

---