Given the set 𝑋 is two, four, the set 𝑌 is four, five, and the universal set 𝑢 is five, four, nine, two, find the complement of the intersection of 𝑋 and the complement of 𝑌.
Before we begin figuring out exactly what this means, let’s draw a Venn diagram to help organize all of our elements. So here we’ve begun to draw the Venn diagram. 𝑋 has elements two and four. And 𝑌 has elements four and five. So 𝑋 and 𝑌 share the element four. And this tiny place where they overlap is where we should put the elements that they have in common. So four will go here. And then 𝑋 also has the element two. And 𝑌 also has the element five.
And the universal set is essentially the entire set. The universal set holds all of the elements. And if there’s another set, it must be a subset of the universal set. So 𝑋 and 𝑌 must be subsets of the universal set. Therefore, they must be within the universal set.
So the universal set has five, four, nine, and two. Five, four, and two are already in 𝑋 and 𝑌. So it’s nine that we need to introduce. And nine is not in 𝑋. And it’s not in 𝑌. So we need to write it outside of these, as we have.
So let’s begin dissecting this. So we have the intersection of 𝑋 and the complement of 𝑌. And we want the complement of the entire thing. So first of all, this symbol means intersection. And the intersection of sets are the elements that are in both sets. So we have the set 𝑋. And then we have the set of the complement of 𝑌.
So the complement of 𝑌, which we know by the line on top of 𝑌, means we want all of the elements that are not in 𝑌. So if we look at the set 𝑌, 𝑌 has elements four and five. So the only other elements that are not in 𝑌 would be nine and two. So the complement of 𝑌 holds the elements nine and two.
And we want the intersection of the complement of 𝑌 and 𝑋. Well, 𝑋 has elements two and four. So if we want the intersection of these two sets, what numbers do they have in common? They have the two in common. So to represent 𝑋 and the intersection of the complement of 𝑌, so just what’s inside the parentheses we have here, it’s just two. Two is the only element that’s in 𝑋 and isn’t in 𝑌.
But now it says that we need to take the complement of that. So we want everything that’s not in that. So all of the other elements that are inside that pink would be nine, four, and five. So this will be our final answer. So the complement of the intersection of 𝑋 and the complement of 𝑌 would be nine, four, and five. And the order that we write these elements does not matter. So once again, this will be our final answer.