Let \(A=\{1,2,3,4\}\) and \(R=\{(a,b): a \text{ divides } b\}\). Write \(R\) Explicitly
Question
Let \[ A=\{1,2,3,4\} \] and \[ R=\{(a,b): a\in A,\ b\in A,\ a \text{ divides } b\}. \] Write \(R\) explicitly.
Solution
Given:
\[ A=\{1,2,3,4\} \]
We form ordered pairs \((a,b)\) such that:
\[ a \mid b \]
that is, \(a\) divides \(b\).
For \(a=1\)
Since 1 divides every element of \(A\),
\[ (1,1),\ (1,2),\ (1,3),\ (1,4) \]
For \(a=2\)
2 divides 2 and 4.
\[ (2,2),\ (2,4) \]
For \(a=3\)
3 divides 3.
\[ (3,3) \]
For \(a=4\)
4 divides 4.
\[ (4,4) \]
Therefore, the relation \(R\) is:
\[ \boxed{ R= \{ (1,1),(1,2),(1,3),(1,4), (2,2),(2,4), (3,3), (4,4) \} } \]