Sets, Relations & Functions
Relations — Counting Elements on Product Set
nta_pyq_2024_apr
Grade 11
Question:
Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A\times B$ by $(a_1,b_1)\,R\,(a_2,b_2)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is _____
Step-by-Step Solution
Key Concept: For each pair of elements $(a_1,b_1),(a_2,b_2)\in A\times B$, check if $a_1+a_2=b_1+b_2$. Enumerate all combinations systematically by the value of $a_1+a_2=b_1+b_2$.
Systematic enumeration gives 24 ordered pairs (from 12 unordered non-trivial matches, each counted twice) plus 1 diagonal pair $(6,6)R(6,6)$. Total $=25$.
Correct Answer: 25