Question Video: Finding a Possible Universal Set given Two of Its Subsets

Video Transcript

Given that 𝑋 is a set containing the elements seven, 11, and 22, 𝑌 is a set containing the elements one, 11, 21, and 22, and 𝑋 and 𝑌 are subsets of some universal set 𝑈, which of the following could be true? 𝑈 is a set containing the element 22, 𝑈 is a set containing the elements one, 11, and 21, 𝑈 is a set containing the elements one, seven, 11, and 21, or 𝑈 is a set containing the elements one, seven, 11, 21, and 22.

This problem tests whether we’re able to say what a possible universal set might contain if we know some of its subsets. The question tells us that the universal set that we’re talking about is 𝑈, and 𝑋 and 𝑌 are the subsets. Let’s represent this as a Venn diagram. Let’s use this green square to represent the universal set 𝑈. This orange circle can represent set 𝑋. Notice that we’ve drawn the circle completely inside the square. This is because 𝑋 is a subset of the universal set 𝑈. Everything in 𝑋 is inside 𝑈. And the numbers in 𝑋 are seven, 11, and 22.

And we can also draw a pink circle to represent set 𝑌, which is also a subset of the universal set. Now, when we look at the elements of set 𝑌, we can see that two of the numbers are also in set 𝑋. So, when we draw our pink circle, we’re going to have to make sure it overlapped with the orange one. So, the two numbers that both sets share are 11 and 22. And then, as we’ve said already, set 𝑋 contains the number seven too, and set 𝑌 contains the numbers one and 21. So, this Venn diagram represents the question.

Now, we’re given four statements about the universal set, 𝑈. And we’re asked which of the following could be true. Each statement gives the letter 𝑈 and then tells us what the elements of that universal set are. So, the correct answer is the one that contains all the elements. When we look inside our green square, we can see the numbers one, seven, 11, 21, and 22. This means that the correct answer is the one that says 𝑈 equals a set containing the elements one, seven, 11, 21, and 22, the five numbers that we can see inside the set.

Notice the word could in the last question. Which of the following could be true? The reason why it could be true, and is not definitely true, is that the question doesn’t tell us that 𝑋 and 𝑌 are the only things within the universal set 𝑈. If they were the only sets within 𝑈, then this answer is correct. But of course, if there was another set there, then we would need to include the numbers in that too. Universal sets contain everything within them.

So, given that 𝑋 is a set containing the elements seven, 11, and 22, and 𝑌 is a set containing the elements one, 11, 21, and 22. And given that both 𝑋 and 𝑌 are subsets of a universal set, which we’ll call 𝑈. The statement that could be true is that 𝑈 is a set containing the elements one, seven, 11, 21, and 22.