Question

Let A be a set. Show that \(\phi \times A = A \times \phi = \phi \).

Short answer

A set with 0 cardinality must be an empty set. It is proved that \(\phi \times A = A \times \phi = \phi \)

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 show that \(\phi  \times A = A \times \phi  = \phi \)\(\)

Here,

The cartesian product of a set is A and B is:

\(A \times B = \left| A \right| \times \left| B \right|\)

Now, if \(A = \phi \) and \(B = \phi \)

Then \(\left| {A \times B} \right| = \left| {B \times A} \right| = 0\)

A set with 0 cardinality must be an empty set.

Therefore, it is proved that \(\phi \times A = A \times \phi = \phi \).