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Question Video: Finding a Set given Its Cartesian Product with Itself Mathematics

If 𝑋² = {(5, 5), (5, 2), (5, 19), (2, 5), (2, 2), (2, 19), (19, 5), (19, 2), (19, 19)}, find 𝑋.

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Video Transcript

If 𝑋 squared is the set of ordered pairs five, five; five, two; five, 19; two, five; two, two; two, 19; 19, five; 19, two; 19, 19, find 𝑋.

We begin by recalling that the Cartesian product 𝐴 times 𝐵 of two sets 𝐴 and 𝐵 is the collection of all ordered pairs 𝑥, 𝑦, where 𝑥 is an element of 𝐴 and 𝑦 is an element of 𝐵. In this question, we are given the set of elements in 𝑋 squared, and we know this is 𝑋 multiplied by 𝑋. Our first three ordered pairs have an 𝑥-value of five. The next three ordered pairs have an 𝑥-value of two. And the final three ordered pairs have an 𝑥-value of 19. This suggests that 𝑋 is the set of numbers five, two, 19.

We can check this by considering the 𝑦-values in the ordered pairs in the set of 𝑋 squared. The number five appears three times. The number two appears three times. And finally, the number 19 also appears three times. This can be represented in the table shown. The Cartesian product of the set with elements five, two, 19, and itself gives the set 𝑋 squared given in the question. This confirms that 𝑋 is the set with elements five, two, and 19.

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