Sets, Relations & Functions
Relations
Grade 11
Question:
<p>The relation R defined on the set of natural numbers as <span class="latex">\{(a, b) : a \text{ differs from } b \text{ by } 3\}</span> is given by</p>
<p>(a) <span class="latex">\{(1, 4), (2, 5), (3, 6), \ldots\}</span></p>
<p>(b) <span class="latex">\{(4, 1), (5, 2), (6, 3), \ldots\}</span></p>
<p>(c) <span class="latex">\{(1, 3), (2, 6), (3, 9), \ldots\}</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: When a differs from b by 3, both |a−b|=3 cases must be included: a=b+3 and a=b−3.
<p><strong>Step 1:</strong> The relation states that <span class="latex">a</span> differs from <span class="latex">b</span> by 3, meaning <span class="latex">|a - b| = 3</span></p><p><strong>Step 2:</strong> This gives two cases:</p><ul><li>Case 1: <span class="latex">a - b = 3 \Rightarrow a = b + 3</span>, giving pairs <span class="latex">\{(4, 1), (5, 2), (6, 3), \ldots\}</span></li><li>Case 2: <span class="latex">b - a = 3 \Rightarrow b = a + 3</span>, giving pairs <span class="latex">\{(1, 4), (2, 5), (3, 6), \ldots\}</span></li></ul><p><strong>Step 3:</strong> The complete relation is the union of both cases: <span class="latex">R = \{(1, 4), (2, 5), (3, 6), \ldots\} \cup \{(4, 1), (5, 2), (6, 3), \ldots\}</span></p><p>∴ Answer is (d) None of these, as both (a) and (b) represent only partial relations.</p>
Correct Answer: d