Question Video: Finding Unknown Elements Using the Equality of Sets

Given that {𝑏, 3, 4} = {𝑐, 9, 4}, find 𝑏 and 𝑐.

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Video Transcript

Given that the set 𝑏, three, and four, is equal to the set 𝑐, nine, and four, find 𝑏 and 𝑐.

This problem compares two sets. They both contain three elements: a letter, which we don’t know the value of, and two numbers. The first set contains the letter 𝑏 and the numbers three and four. The second set contains another letter 𝑐 and this time the numbers nine and four. And, the equal sign in between both sets tells us that they’re both the same. But how can they be equal with each other when they contain different letters, and some of the numbers are different, too? The question asks us to find the value of 𝑏 and 𝑐.

But to solve the problem, to begin with, we need to ignore 𝑏 and 𝑐 and just look at the numbers that are already in each set. We can use what we know already to help us find the value of the letters. So what do we know about the numbers in each set? We know one of the numbers is the number four. This number appears in both of the sets in the question.

We can also see that in the first set, there is the number three. We can’t see it in the second set, but we know it has to be there because the second set is equal to the first set. So where’s the number three in the second set? The letter 𝑐 must be equal to three. In fact, we could cross out the letter 𝑐 and replace it with the number three.

We can use the same reasoning to find the value of 𝑏. In the second set, we can see the number nine. And, the number nine isn’t in the first set at all. But, we know the number nine should appear in the first set because it’s equal to the second set. The only thing we can say is that 𝑏 must equal nine. And so, we can cross out 𝑏 and replace it with the number nine.

If 𝑏 is worth nine and 𝑐 is worth three, then look at each set. They’re exactly the same. They contain the elements four, three, and nine. So if the set 𝑏, three, and four is equal to the set 𝑐, nine, and four, then 𝑏 must equal nine and 𝑐 must equal three. This way, the three elements in both sets are exactly the same.

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