Question

Write out the indicated sets by listing their elements between braces. $$ \\{\varnothing\\} \times\\{0, \varnothing\\} \times\\{0,1\\} $$

Short answer

The product \({\varnothing}\) \(\times\) \({0, \varnothing}\) \(\times\) \({0,1}\) equals \({\varnothing}\). The presence of the empty set in the Cartesian product causes the entire product to be an empty set.

Step-by-step solution

Analyze the empty set

The empty set, denoted by \({\varnothing}\), contains no elements. Therefore, the product of \({\varnothing}\) with any other set is always \({\varnothing}\), since there are no elements to be multiplied.

Compute \({\varnothing}\) \(\times\) \({0, \varnothing}\)

Applying the rule stated in Step 1, the product of \({\varnothing}\) and any set including \({0, \varnothing}\) is \({\varnothing}\). So, \({\varnothing}\) \(\times\) \({0, \varnothing}\) = \({\varnothing}\).

Compute \({\varnothing}\) \(\times\) \({0,1}\)

Following the rule from Step 1, since the set on the left-hand side is \({\varnothing}\), any product with another set, in this case \({0,1}\), results in \({\varnothing}\). So, \({\varnothing}\) \(\times\) \({0,1}\) = \({\varnothing}\).

Putting it all together

Now, we combine the results from steps 2 and 3. The original expression \({\varnothing}\) \(\times\) \({0, \varnothing}\) \(\times\) \({0,1}\) simplifies to \({\varnothing}\) \(\times\) \({\varnothing}\) = \({\varnothing}\).