Video: Finding the Union of a Set and the Complement of Another Using Venn Diagrams

Using the Venn diagram, find π‘Œ βˆͺ 𝑋 bar.

02:09

Video Transcript

Using the Venn diagram, find π‘Œ union 𝑋 bar.

Let’s begin by defining some of the notation in this question. The U-shaped symbol between π‘Œ and 𝑋 bar means the union. In probability, this means or. We need to find the set of values that are in π‘Œ or in 𝑋 bar. The notation 𝑋 bar means the complement of 𝑋. This is the set of all values not in 𝑋.

We can see from the Venn diagram that set π‘ˆ is all the numbers shown. This is the integer values from one to nine. The set of values contained in circle π‘Œ are one, two, six, and nine. The set of values in circle 𝑋 are one, three, and six. This means that the complement of 𝑋 will be all the values except these. The numbers that are not in set 𝑋 are two, four, five, seven, eight, and nine.

We recall that we need to find the values that are in set π‘Œ or in set 𝑋 bar. The numbers two and nine appear in both of these sets, but we do not need to write this number twice. The set of numbers contained within the union of π‘Œ and 𝑋 bar are one, two, four, five, six, seven, eight, and nine. In fact, the only number that is not in this union is three. This is because it appears in set 𝑋, but not in set π‘Œ. As we want all the values that are either in set π‘Œ or not in set 𝑋, we do not include the three.

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