Use the given Venn diagram to find the complement of 𝑋 union 𝑌 union 𝑍.
Well, in this question, the first thing we wanna take a look at is the notation and what it means. Well, we can see that we’ve got 𝑋 union 𝑌 union 𝑍. So we have this U, and what this U means is, as I’ve said, union. But what this can also mean is or. So we could think of it as 𝑋 or 𝑌. So, for instance, in our example, we’d have 𝑋 or 𝑌 or 𝑍. Then the other bit of notation we’ve got is here a bar. And this bar is on top. So in our example, we’ve got the bar above the 𝑋 union 𝑌 union 𝑍. But what does this mean? Well, if we’ve got 𝑋 bar and this means the complements of 𝑋, so this means not 𝑋. So we’re gonna be looking for the set of values that are not in the set 𝑋.
So therefore, for this question, we could think of the complement of 𝑋 union 𝑌 union 𝑍 as the set of values that are not in 𝑋 or 𝑌 or 𝑍. So, first of all, if we’d look at the set of values that are in 𝑋 union 𝑌 union 𝑍, then these would be zero, one, two, three, five, seven, nine, 10, 11. And I’ve circled them in pink here. So therefore, if you want to find the complement of 𝑋 union 𝑌 union 𝑍, so, as we’ve said, the set of values not in 𝑋 or 𝑌 or 𝑍, then this is gonna be the set eight, six.