Question Video: Finding the Union of the Complements of Two Sets Using a Given Venn Diagram

Video Transcript

Using the Venn diagram, find the union of 𝑋 bar and 𝑌 bar.

Let’s begin by defining some of the notation in this question. The U or union symbol means “or.” This means we’re looking for the set of values that are in 𝑋 bar or in 𝑌 bar. 𝑋 bar means the complement of 𝑋. This is all the values that are not in set 𝑋. Likewise, 𝑌 bar or the complement of 𝑌 is the set of values not in 𝑌. We need to find the set of values that are not in 𝑋 or not in 𝑌.

We can see from the Venn diagram that 𝑋 contains the set of values three, six, seven, and nine. The complement of 𝑋 or those values not in 𝑋 will be all the other values, in this case, one, two, four, five, and eight. The numbers in set 𝑌 are one, three, five, and nine. The numbers not in set 𝑌 or the complement of 𝑌 are two, four, six, seven, and eight. We need to select all the numbers that are either not in 𝑋 or not in 𝑌.

Two, four, and eight appear in both of these lists. But we do not need to write them twice. The set of numbers that appear in the complement of 𝑋 or the complement of 𝑌 are one, two, four, five, six, seven, and eight. This is actually all of the numbers except three and nine. These are the numbers that appear in both 𝑋 and 𝑌. We can therefore conclude that the union of the complement of 𝑋 and the complement of 𝑌 is everything apart from the intersection of 𝑋 and 𝑌.