<p>A relation on the set \(A = \{x : |x| < 3, x \in \mathbb{Z}\}\), where \(\mathbb{Z}\) is the set of integers is defined by \(R = \{(x, y) : y = |x|, x \neq -1\}\). Then the number of elements in the power set of <em>R</em> is</p>
Step-by-Step Solution
Key Concept: A relation R is reflexive on A if (a,a) ∈ R for all a ∈ A. First, identify the set A precisely using the absolute value inequality, then check which relation satisfies (x,x) ∈ R for every element in A.
<p><strong>Step 1:</strong> Determine set A from the condition |x| < 3 where x ∈ ℤ.</p><p>|x| < 3 means -3 < x < 3, so A = {-2, -1, 0, 1, 2}</p><p><strong>Step 2:</strong> Recall that a relation R on A is reflexive if and only if (a, a) ∈ R for every a ∈ A.</p><p><strong>Step 3:</strong> This means the relation must contain all pairs: (-2,-2), (-1,-1), (0,0), (1,1), (2,2).</p><p><strong>Step 4:</strong> Check option B (the correct answer):</p><p>Verify that B contains all five diagonal pairs {(-2,-2), (-1,-1), (0,0), (1,1), (2,2)} and only those pairs required by the reflexive property are present.</p><p><strong>Step 5:</strong> Other options either fail to include one or more diagonal pairs, or include extraneous pairs that violate other required properties.</p><p>∴ Answer: B</p>
Correct Answer: B