<p>If <math>I</math> is the set of integers and if the relation <math>R</math> is defined over <math>I</math> by <math>aRb</math>, iff <math>a - b</math> is an even integer, <math>a, b \in I</math>, the relation <math>R</math> is</p>
Step-by-Step Solution
Key Concept: Check reflexivity, symmetry, and transitivity for a relation. A relation is an equivalence relation if and only if it is reflexive, symmetric, and transitive.
<p><strong>Reflexivity:</strong> <math>aRa \Rightarrow a - a = 0</math> (even integer). Thus <math>(a, a) \in R, \forall a \in I</math>. So <math>R</math> is reflexive.</p><p><strong>Symmetry:</strong> Let <math>(a, b) \in R \Rightarrow (a - b)</math> is an even integer. Then <math>-(b - a)</math> is an even integer, so <math>(b - a)</math> is an even integer. Thus <math>(b, a) \in R</math>. So <math>R</math> is symmetric.</p><p><strong>Transitivity:</strong> Let <math>(a, b) \in R</math> and <math>(b, c) \in R</math>. Then <math>a - b = 2x_1</math> and <math>b - c = 2x_2</math> for some <math>x_1, x_2 \in I</math>. Thus <math>a - c = 2(x_1 + x_2)</math> is even, so <math>(a, c) \in R</math>. So <math>R</math> is transitive.</p><p>Hence, <math>R</math> is an equivalence relation.</p>
Correct Answer: a, c, d