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Cartesian Product with the Empty Set
publish date: 2026/05/23 22:20:49.621781 UTC
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Let \(A = \{1, 2, 3\}\) and \(B = \varnothing\). What is \(A \times B\)?
Correct Answer
\(\varnothing\)
Explanation
The Cartesian product \(A \times B\) is the set of all ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\). If \(B = \varnothing\), there is no possible choice for \(b\). Therefore, there are no ordered pairs, and \(A \times B = \varnothing\). The same holds if \(A = \varnothing\) or if both are empty.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012