Relations & Functions
Composition of Relations
Grade 12
Question:
<p>If the relation <i>R</i> : <i>A</i> → <i>B</i>, where <i>A</i> = {1, 2, 3, 4} and <i>B</i> = {1, 3, 5} is defined by <i>R</i> = {(<i>x</i>, <i>y</i>) : <i>x</i> < <i>y</i>, <i>x</i> ∈ <i>A</i>, <i>y</i> ∈ <i>B</i>}, then <i>RoR</i><sup>−1</sup> is</p>
<p>(a) {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}</p>
<p>(b) {(3, 1), (5, 1), (5, 2), (5, 3), (5, 4)}</p>
<p>(c) {(3, 3), (3, 5), (5, 3), (5, 5)}</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: To find the composition RoR⁻¹, first find R⁻¹ by reversing all pairs in R, then compose by matching endpoints of pairs from R⁻¹ and R.
<p><strong>Step 1:</strong> Find <i>R</i> by the condition <i>x</i> < <i>y</i>, <i>x</i> ∈ <i>A</i>, <i>y</i> ∈ <i>B</i>:</p><p><i>R</i> = {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}</p><p><strong>Step 2:</strong> Find <i>R</i><sup>−1</sup> by swapping elements:</p><p><i>R</i><sup>−1</sup> = {(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}</p><p><strong>Step 3:</strong> Compute <i>RoR</i><sup>−1</sup> by finding pairs (<i>a</i>, <i>c</i>) where (<i>a</i>, <i>b</i>) ∈ <i>R</i><sup>−1</sup> and (<i>b</i>, <i>c</i>) ∈ <i>R</i>:</p><p><i>RoR</i><sup>−1</sup> = {(3, 3), (3, 5), (5, 3), (5, 5)}</p><p>∴ Answer is (c).</p>
Correct Answer: C