Question Video: Using Venn Diagrams to Find the Difference of the Complements of Two Sets

Using the Venn diagram, find ๐‘‹โ€ฒ โˆ’ ๐‘Œโ€ฒ.

04:05

Video Transcript

Using the Venn diagram, find the complement of ๐‘‹ minus the complement of ๐‘Œ.

Now, when I read the question out, I read this notation automatically. ๐‘‹ and then a little superscript dash means the complement of the set ๐‘‹. The complement of ๐‘‹ is the set of all elements which are in the universal set, so thatโ€™s all the elements weโ€™re interested in in this question, but arenโ€™t in the set ๐‘‹. We can write this as the universal set ๐‘ˆ minus the set ๐‘‹ if we wish. The same is true for the notation ๐‘Œ and then a dash. It means the complement of ๐‘Œ. So the set of all elements which are in our universal set ๐‘ˆ but arenโ€™t in the set ๐‘Œ.

Letโ€™s use the Venn diagram to find each of these sets individually. For ๐‘‹ prime then, weโ€™re looking for all of the elements which are inside the universal set. So theyโ€™re in the grey box. But they arenโ€™t in set ๐‘‹. So they arenโ€™t inside the pink oval. There are five elements of this set, the elements one and five, which are in neither ๐‘‹ or ๐‘Œ, but are in the universal set and the elements zero, three, and eight, which are in set ๐‘Œ, but not in set ๐‘‹. So the complement of ๐‘‹ is the set containing the elements zero, one, three, five, and eight.

Now, we can look for the set which is the complement of ๐‘Œ. Remember, this is all the elements which are in the universal set. So they are within the grey box. But they arenโ€™t in set ๐‘Œ. So theyโ€™re not within the blue oval. We can see that this set consists of three elements, the elements one and five, which are in the universal set but are not in either ๐‘‹ or ๐‘Œ, and the element six, which is in set ๐‘‹ but isnโ€™t in set ๐‘Œ. Itโ€™s in the pink oval, but not the blue oval.

To find the set, the complement of ๐‘‹ minus the complement of ๐‘Œ, we need to take the set zero, one, three, five, eight and then subtract the elements one, five, and six. Letโ€™s have a look then. So the element zero is in the complement of ๐‘‹. But it isnโ€™t in the complement of ๐‘Œ. So it will be in the complement of ๐‘‹ minus the complement of ๐‘Œ. The element one is in both of these sets, which means it isnโ€™t in the complement of ๐‘‹ minus the complement of ๐‘Œ. The element three is in the complement of ๐‘‹. But it isnโ€™t in the complement of ๐‘Œ. So weโ€™re not subtracting it off, which means that the element three will be in this set.

The element five is in both sets, which means we will be taking this element out. So we do not have it in the complement of ๐‘‹ minus the complement of ๐‘Œ. The element eight is in the complement of ๐‘‹ but isnโ€™t in the complement of ๐‘Œ. So we keep hold of this element. And finally, there is one element in the complement of ๐‘Œ, which we havenโ€™t mentioned. It is the element six. But remember, weโ€™re starting with the elements which are in the complement of ๐‘‹. So our starting set of values is zero, one, three, five, and eight. And then, weโ€™re just taking out of this set the elements that also appear in the complement of ๐‘Œ. As the element six didnโ€™t appear in our starting set, it makes no difference if we try to take it out.

Weโ€™re left then with three elements, which are in the complement of ๐‘‹ but arenโ€™t in the complement of ๐‘Œ. Those elements are zero, three, and eight. We can also see this directly on our Venn diagram. The elements zero, three, and eight are elements which we underlined in orange but didnโ€™t underline in pink. So theyโ€™re elements that are in the complement of ๐‘‹ but arenโ€™t in the complement of ๐‘Œ.

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