Write R as a Set of Ordered Pairs for |a² − b²| ≤ 5
Question
Let
\[ A=\{1,2,3\} \]
and
\[ R=\{(a,b): |a^2-b^2|\le5,\ a,b\in A\} \]
Then write \( R \) as a set of ordered pairs.
Solution
\[ |1^2-1^2|=0\le5 \]
\[ |1^2-2^2|=3\le5 \]
\[ |1^2-3^2|=8>5 \]
\[ |2^2-1^2|=3\le5 \]
\[ |2^2-2^2|=0\le5 \]
\[ |2^2-3^2|=5\le5 \]
\[ |3^2-1^2|=8>5 \]
\[ |3^2-2^2|=5\le5 \]
\[ |3^2-3^2|=0\le5 \]
Therefore,
\[ R=\{(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3)\} \]
Hence,
\[ \boxed{\{(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3)\}} \]