12) Determine whether or not the following is an equivalence relation on the set of all integers. Number a is related to number \( \mathrm{b}(a R b) \) if \( a-b \) is even. Determine whether or not this is an equivalence relation. If it is, prove it. If it is not, show how it fails to be an equivalence relation. ( 7 points)

To determine if the relation \( R \) defined by \( a R b \) if \( a - b \) is even constitutes an equivalence relation, we need to check three properties: reflexivity, symmetry, and transitivity. 1. **Reflexivity**: For any integer \( a \), \( a - a = 0 \), which is even. Thus, \( a R a \) holds for all integers \( a \). Reflexivity is satisfied! 2. **Symmetry**: If \( a R b \), then \( a - b \) is even. This means \( b - a = -(a - b) \), which is also even (the negation of an even number is even). Therefore, if \( a R b \), then \( b R a \). Symmetry is satisfied! 3. **Transitivity**: If \( a R b \) (meaning \( a - b \) is even) and \( b R c \) (meaning \( b - c \) is even), we can add these: \( (a - b) + (b - c) = a - c \), which is the sum of two even numbers, hence even. Thus, if \( a R b \) and \( b R c \), then \( a R c \) as well. Transitivity is satisfied! Since \( R \) satisfies reflexivity, symmetry, and transitivity, we conclude that the relation is indeed an equivalence relation on the set of all integers. To put a cherry on top: you could visualize this in terms of even and odd integers. All even integers are related to one another, and the same goes for odd integers. It’s like having exclusive parties — even numbers hang out with even numbers, while odd numbers party with odd ones!