Question Video: Using Venn Diagrams to Represent Sets given Their Complements and the Universal Set

Video Transcript

Take the universal set to be the set of numbers between, but not including, one and 10. Given that the complement of set 𝑋 equals seven, three, eight and the complement of set 𝑌 equals four, nine, three, which Venn diagram represents these sets?

So in order to choose which one of our Venn diagrams is in fact the correct one, what we’re gonna look at first is a complement of 𝑋, so the complement of set 𝑋 being equal to seven, three, and eight. What this means is that not 𝑋 is seven, three, and eight. So seven, three, and eight are not in a set of 𝑋. Well, we can see that the Venn diagram (b) does in fact have seven, three, and eight inside of the set 𝑋. So therefore, this cannot be the correct Venn diagram. But if we look at any other Venn diagrams, we can see that seven, three, and eight are not in the set of 𝑋. So that’s the only thing that we can use or ascertain from the fact that we know that the complement of set is 𝑋 seven, three, and eight.

So now, what we need to do is move on to the fact that the complement of set 𝑌 is equal to four, nine, and three. So this means that 𝑌 cannot be four, nine, or three. Well, on the face of it, we can think, “well, this only rules out (b) again because this is the only one in which 𝑌, so the set 𝑌, contains four, nine, or three.” However, if we look closely at the two bits of information we’ve been given together, we can see that, for instance, the number three is not in set 𝑋, but also not in set 𝑌. So therefore, we can also rule out our Venn diagram (d) because this doesn’t in fact even have a number three anywhere on the Venn diagram.

Well, if you take a look at Venn diagram (a) and Venn diagram (c), these both, in fact, have three that’s not in set 𝑋 or not in set 𝑌. However, we can rule out Venn diagram (c) as well because, okay, it’s got the three outside of the sets 𝑋 and 𝑌. However, if we look at the information we’ve been given, we’ve been told that the only value or the only number that isn’t in set 𝑋 and isn’t in set 𝑌 is three. However, in (c), so Venn diagram (c), we can also see six, two, and five would fall into this category. So therefore, (c) is not correct. So therefore, we can say that the Venn diagram, which correctly represents our sets, is in fact Venn diagram (a).