<p>Let \(T = \{(x, y) : x - y \in I\}\). Then which of the following is true?</p>
<p><em>T</em> is an equivalence relation on the real numbers</p>
<p><em>T</em> is reflexive but not symmetric</p>
<p><em>T</em> is symmetric but not transitive</p>
<p><em>T</em> is transitive but not reflexive</p>
Step-by-Step Solution
Key Concept: T represents all ordered pairs where the difference x - y is an integer. This is a relation on ℝ × ℝ that is reflexive (x - x = 0 ∈ I), symmetric (if x - y ∈ I, then y - x = -(x - y) ∈ I), and transitive (if x - y ∈ I and y - z ∈ I, then x - z = (x - y) + (y - z) ∈ I). Therefore, T is an equivalence relation that partitions ℝ into equivalence classes of the form [a] = {x ∈ ℝ : x - a ∈ I}.
<p><strong>Step 1:</strong> Identify what T represents: T = {(x, y) : x - y ∈ I} where I is the set of integers.</p><p><strong>Step 2:</strong> Check reflexivity: For any x ∈ ℝ, x - x = 0 ∈ I, so (x, x) ∈ T. ✓</p><p><strong>Step 3:</strong> Check symmetry: If (x, y) ∈ T, then x - y ∈ I. Since I is closed under negation, -(x - y) = y - x ∈ I, so (y, x) ∈ T. ✓</p><p><strong>Step 4:</strong> Check transitivity: If (x, y) ∈ T and (y, z) ∈ T, then x - y ∈ I and y - z ∈ I. Since I is closed under addition, (x - y) + (y - z) = x - z ∈ I, so (x, z) ∈ T. ✓</p><p><strong>Step 5:</strong> Since T satisfies all three properties, T is an equivalence relation on ℝ. It partitions ℝ into equivalence classes where two reals are equivalent if and only if their difference is an integer.</p><p>∴ Answer: A (T is an equivalence relation)</p>
Correct Answer: A