Use the given Venn diagram to find the union of 𝑋 bar intersection 𝑌 and 𝑌 bar intersection 𝑍.
Let’s begin by defining some of the notation in this question. The little n notation means the intersection or “and.” We need to find those values that are in 𝑋 bar and 𝑌 and then, separately, those values in 𝑌 bar and 𝑍. The u notation means the union or “or.” We need to find those numbers that are in either one of the sets inside the parentheses or brackets.
The 𝑋 bar and 𝑌 bar notation mean the complement of 𝑋 and the complement of 𝑌. These are the values that are not in set 𝑋 and not in set 𝑌, respectively. Let’s begin by considering the first set of parentheses or brackets. The set of values in 𝑋 bar are all the numbers except for five, seven, and eight, as these lie in set 𝑋. The complement of 𝑋 is therefore equal to one, two, three, four, six, and nine. Set 𝑌 contains the values three, four, five, and six. We are looking for values that are in both of these sets, the intersection. As three, four, and six appear in both sets, the intersection of 𝑋 bar and 𝑌 is three, four, six.
We can repeat this process for our second set of parentheses. Five values are not in set 𝑌. Therefore, the complement of 𝑌 is equal to one, two, seven, eight, and nine. Set 𝑍 contains the numbers three, four, seven, and nine. Once again, we need to find the intersection of these two sets. The numbers seven and nine appear in both these sets. Therefore, the intersection of 𝑌 bar and 𝑍 is seven and nine. Our final step is to find the union of these two sets. As the union means or, we need to find all the numbers that are either in the set 𝑋 bar intersection 𝑌 or the set 𝑌 bar intersection 𝑍. This is the set of values three, four, six, seven, and nine.