Sets, Relations & Functions
General
Grade 11
Question:
<p>Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation on A × B defined by
(a1, b1), (a2, b2)
∈R iff a1 ≤b2 and b1 ≤a2. The number of elements in R is:</p>
<p>26</p>
<p>160</p>
<p>180</p>
<p>52</p>
Step-by-Step Solution
Key Concept: For fixed (a1, b1): count #{a2 \in A : a2 \geqb1} \times #{b2 \in B : b2 \geqa1}. The AND condition separates
— counts are independent. Sum over all 25 pairs.
<p><strong>Step 1</strong>: Count functions: #{b2 \in B : b2 \geqa1}: a1=1 \to 5; a1=3 \to 4; a1=4 \to 4; a1=6 \to 2; a1=9 \to 1.</p><br>#{a2 \in A : a2 \geqb1}: b1=2 \to 4; b1=4 \to 3; b1=5 \to 2; b1=8 \to 1; b1=10 \to 0.<p><strong>Step 2</strong>: Sum products across the 5 \times 5 grid:</p><br>a1\b1<br>2<br>4<br>5<br>8<br>10<br>Row sum<br>1<br>20<br>15<br>10<br>5<br>0<br>50<br>3<br>16<br>12<br>8<br>4<br>0<br>40<br>4<br>16<br>12<br>8<br>4<br>0<br>40<br>6<br>8<br>6<br>4<br>2<br>0<br>20<br>9<br>4<br>3<br>2<br>1<br>0<br>10<br>Total<br>160
Correct Answer: 2