Video: Finding an Unknown Element Using the Relation between Two Given Sets

Given that {8, 7} ⊂ {9, 8, 𝑥}, find the value of 𝑥.

03:10

Video Transcript

Given that a set containing the elements eight and seven is a subset of a set containing the elements nine, eight, and 𝑥, find the value of 𝑥.

In this problem, we’ve got two sets that are being described. Let’s try and represent them using circles. The first set contains the elements eight and seven. And we can see them written there in between the braces with a comma in between them. Our second set contains three elements: nine, eight, and the letter 𝑥, which we know represents a number. We know this because often in algebra, the letter 𝑥 represents numbers. But also in this particular question, we’re asked to find the value of 𝑥.

So, here are our two sets. What relationship do they have to each other? Well, in between the notation for both sets, we have this symbol. Let’s remind ourselves what it means. Well, this symbol, which looks a bit like a stretched-out letter C, means is a subset of. Now, when a set is a subset of another set, it’s contained within it. So, if we said A is a subset of B, it’s contained within B. It’s part of B.

Let’s have another look at the sets in our question. We’re told that a set containing the elements eight and seven is a subset of, or is part of, a set containing the elements nine, eight, and 𝑥. Let’s try drawing our circles again to reflect this. This time, we can draw our second set first. It contains the elements, remember, nine, eight, and 𝑥.

But we’re told that a set containing the elements eight and seven is a subset of this. Well, we should have no problem understanding where the number eight comes from. We can see that in the second set. But where is the number seven? How could we make a subset containing eight and seven when all we have are the digits nine, eight, and of course the letter 𝑥?

Well, the answer is, of course, that the letter 𝑥 must have a value. And we can see what this value is because we know what the subset of this set is. Let’s draw a circle within this set to show the subset. So, now we can write that a set containing the elements eight and 𝑥 is a subset of a set containing the elements nine, eight, and 𝑥. And so, if 𝑥 equals seven, then the statement in the question is still true. If a set containing the elements eight and seven is a subset of a set containing the elements nine, eight, and 𝑥, then we can say that 𝑥 equals seven.

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