Question Video: Identifying Which Cartesian Product of Two Given Sets Would Contain a Given Element Mathematics

Given that 𝑋 = {0, −1} and 𝑌 = {8, −4, −3, −2}, then which of the following relations would (−1, −4) be an element of? [A] 𝑋² [B] 𝑌² [C] 𝑋 × 𝑌 [D] 𝑌 × 𝑋

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Video Transcript

Given that 𝑋 is equal to the set zero, negative one and 𝑌 is equal to the set eight, negative four, negative three, negative two, then which of the following relations would negative one, negative four be an element of? Is it option (A) 𝑋 squared, (B) 𝑌 squared, (C) 𝑋 times 𝑌, or (D) 𝑌 times 𝑋?

The times notation in options (C) and (D) corresponds to the Cartesian product of sets 𝑋 and 𝑌 and sets 𝑌 and 𝑋, respectively. The Cartesian product is the set of all ordered pairs. Before starting this question, it is worth noting that option (A) 𝑋 squared is the same as the Cartesian product of set 𝑋 and set 𝑋. Likewise, 𝑌 squared is equal to the Cartesian product of set 𝑌 and set 𝑌.

The first value of our ordered pair, negative one, occurs in set 𝑋. The second value, negative four, occurs in set 𝑌. We can therefore conclude that option (C) the Cartesian product of 𝑋 and 𝑌 is the correct answer. This relation contains the ordered pair negative one, negative four.

An alternative method here would be to list all the ordered pairs of each relation. For example, the Cartesian product of 𝑋 and 𝑌 contains the ordered pairs zero, eight; zero, negative four; zero, negative three; zero, negative two; negative one, eight; negative one, negative four; negative one, negative three; and negative one, negative two. Once again, we see that negative one, negative four is contained in this Cartesian product.

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