If \(n(A)=3\) and \(n(B)=2\), Find Sets \(A\) and \(B\) from Ordered Pairs in \(A\times B\)
Question
Let \(A\) and \(B\) be two sets such that
\[ n(A)=3 \]
and
\[ n(B)=2. \]
If \[ (x,1),\ (y,2),\ (z,1) \] are in \[ A\times B, \] find the sets \(A\) and \(B\), where \(x,\ y,\ z\) are distinct elements.
Solution
Since \[ (x,1),\ (y,2),\ (z,1)\in A\times B, \] the first elements belong to set \(A\) and the second elements belong to set \(B\).
Elements of Set \(A\)
From the ordered pairs:
\[ (x,1),\ (y,2),\ (z,1) \]
the first components are:
\[ x,\ y,\ z \]
Since \(x,\ y,\ z\) are distinct elements and \[ n(A)=3, \] therefore:
\[ \boxed{A=\{x,y,z\}} \]
Elements of Set \(B\)
The second components are:
\[ 1,\ 2,\ 1 \]
Distinct elements are:
\[ 1,\ 2 \]
Since \[ n(B)=2, \] therefore:
\[ \boxed{B=\{1,2\}} \]
Hence, the required sets are:
\[ \boxed{A=\{x,y,z\}} \]
and
\[ \boxed{B=\{1,2\}} \]