Use the Venn diagram below to find the intersection of 𝑍 bar with the intersection of 𝑋 bar and 𝑌 bar.
Let’s begin by recalling what we mean by the notation in the question. The n symbol means the intersection or “and.” For the part of our question in the parentheses or brackets, we need to find the set of numbers that are in 𝑋 bar and in 𝑌 bar. The notation 𝑋 bar, sometimes written as 𝑋 prime, means the complement of 𝑋. This is the set of all values that are not in 𝑋.
We can see from the Venn diagram that the three values zero, three, and six lie in set 𝑋. This means that 𝑋 bar or the complement of 𝑋 contains all the other values in the Venn diagram: one, two, four, five, seven, and nine. Note that the number eight does not appear inside this Venn diagram. The set of values contained within 𝑌 are zero, one, four, and five. This means that 𝑌 bar, or the complement of 𝑌, contains two, three, six, seven, and nine.
The set of values in the intersection of 𝑋 bar and 𝑌 bar will be contained in both of these sets. There are three such numbers: two, seven, and nine. These are the three numbers that are not in set 𝑋 and also not in set 𝑌.
Set 𝑍 contains the four numbers one, four, six, and seven. This means that the complement of 𝑍 is equal to zero, two, three, five, and nine. We need to find the numbers that are in this set and also in the set of the intersection of 𝑋 bar and 𝑌 bar. The numbers two and nine exist in both of these sets.
The intersection of 𝑍 bar with the intersection of 𝑋 bar and 𝑌 bar is two and nine. These are the two values that did not exist in any of the three sets. They’re not in 𝑋, 𝑌, or 𝑍.