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Question Video: Finding the Cartesian Product of a Set and an Intersection of Sets Mathematics

If 𝑋 = {8, 4, 6}, 𝑌 = {6, 7}, and 𝑍 = {7}, find 𝑋 × (𝑌 ⋂ 𝑍).

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Video Transcript

If set 𝑋 contains the elements eight, four, six; set 𝑌 is equal to six, seven; and set 𝑍 is equal to seven, find the Cartesian product of set 𝑋 and the intersection of set 𝑌 and 𝑍.

We recall that the Cartesian product of two sets is the collection of all ordered pairs. The intersection of two sets is those elements that occur in both sets, in this case in set 𝑌 and 𝑍. Set 𝑌 contains the elements six and seven, whereas set 𝑍 only contains the number seven. This means that the only number that appears in both sets is seven. The intersection of set 𝑌 and 𝑍 is equal to seven. We are told in the question that set 𝑋 contains the numbers eight, four, and six.

We now need to find the Cartesian product of these two sets. As our first value in the Cartesian product is 𝑋, each of the elements of 𝑋 will be the first number in the ordered pairs. Our first ordered pair is therefore eight, seven. Next, we have four, seven. Finally, we have the ordered pair six, seven. Each of our values in set 𝑋 has now been matched with the value of seven in the set 𝑌 intersection 𝑍. The Cartesian product contains the ordered pairs eight, seven; four, seven; and six, seven.

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