Question

Suppose that \(A \times B = \phi \) , where A and B are sets. What can you conclude?

Short answer

\(A = \phi \) or \(B = \phi \)

Step-by-step solution

Cartesian product and Empty set

The cartesian product of A and B denoted by\({\bf{A \times B}}\), is the set of all ordered pairs (a,b).

Therefore, \(A \times B = \left\{ {\left( {a,b} \right)\left| {a \in A\Lambda b \in B} \right.} \right\}\)

The empty set does not contain any element.

The \(\phi \) denotes the empty set.

To draw the conclusion from given statement

Here,

\(A \times B = \phi \)

By the definition of Cartesian Product, \(A \times B\) does not contain any element and also \(a \in A\) and \(b \in B\).

Therefore, it means that either A does not contain any element or B does not contain any element.

Thus, \(A = \phi \) or \(B = \phi \).