State Whether the Statement is True or False: If \(A\) and \(B\) are Non-Empty Sets, Then \(A\times B\) is a Set of Ordered Pairs \((x,y)\) Such That \(x\in B\) and \(y\in A\)
Question
State whether the following statement is true or false. If the statement is false, rewrite the statement correctly:
If \(A\) and \(B\) are non-empty sets, then \[ A\times B \] is a non-empty set of ordered pairs \((x,y)\) such that \[ x\in B \] and \[ y\in A. \]
Solution
The given statement is:
\[ \boxed{\text{False}} \]
In the Cartesian product \[ A\times B, \] the first element of the ordered pair belongs to set \(A\) and the second element belongs to set \(B\).
That is,
\[ (x,y)\in A\times B \] such that
\[ x\in A \] and
\[ y\in B. \]
Correct Statement
If \(A\) and \(B\) are non-empty sets, then \[ A\times B \] is a non-empty set of ordered pairs \((x,y)\) such that
\[ x\in A \]
and
\[ y\in B. \]
Therefore,
\[ \boxed{ A\times B=\{(x,y):x\in A,\ y\in B\} } \]