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Video: Verifying the Relations between the Cartesian Products of Two Given Sets

Bethani Gasparine

If 𝑋 = {37, 18, 39} and 𝑌 = {9, 12}, is (𝑌 × 𝑌) ⊂ (𝑋 × 𝑋)?

02:57

Video Transcript

If 𝑋 is the set of elements 37, 18, and 39 and 𝑌 is the set of elements nine and 12, is the product of 𝑌 times 𝑌 a subset of the product of 𝑋 times 𝑋?

The product of the 𝑌s contains ordered pairs made from the elements of 𝑌, and the product of the 𝑋s contain ordered pairs made from the elements of 𝑋. And none of the elements of 𝑌 are in 𝑋.

Well, let’s begin by looking at the products. In order to create this set of the product of 𝑌 and 𝑌, we need to use the elements of 𝑌. So the elements of 𝑌 are nine and 12. So we need to create ordered pairs using these elements.

So let’s begin with using the nine. And we can pair nine with itself or 12. So nine, nine would be an ordered pair and nine, 12 would be an ordered pair. Now let’s begin with 12. We can either pair 12 with itself or with nine. So we have 12, nine and 12, 12. And this would complete the product of 𝑌 and 𝑌.

And now for the 𝑋s. 𝑋 has elements 37, 18, and 39. So let’s begin with 37. 37 can be paired with itself, paired with 18, or paired with 39. Now let’s look at 18 as the first of the ordered pair. So 18 could be paired with 37. It could be paired with itself or paired with 39. Now with 39, it can be paired with 37. It could be paired with 18 or it could be paired with itself.

So the question is asking if the product of the 𝑌s is a subset of the product of the 𝑋s, so this means subset. So in order for the part of the 𝑌s to be a subset of the product of the 𝑋s, all of the elements of 𝑌 times 𝑌 must also be found in the product of the 𝑋s.

So we have nine, nine for the 𝑌 product. And that is not found on the 𝑋 products. Nine, 12 is also not found. So what about 12, nine? That’s not gonna work because it’s not found in the set of all the blue ordered pairs, and neither is 12, 12.

And like we said before, this is because the elements that are found in 𝑌 originally, nine and 12, they are not found in the set of 𝑋, which means it wouldn’t change for the products. Therefore, answer will be no.