Question Video: Finding the Union of Two Cartesian Products of Given Sets Mathematics

Video Transcript

If the set 𝑋 is equal to eight, 𝑌 is equal to eight, three, and 𝑍 is equal to nine, four, five, find the union of the Cartesian products of 𝑋 and 𝑌 and 𝑌 and 𝑍.

So to solve this problem, what we need to do is first find the products of 𝑋 and 𝑌 and 𝑌 and 𝑍, and then find their union. So I’m gonna start with 𝑋 multiplied by 𝑌, or the product of 𝑋 and 𝑌. So to find the Cartesian product, so we multiply 𝑋 and 𝑌, what we need to do is write down a set of ordered pairs with the coordinates 𝑋, 𝑌 that come from 𝑋 being an element of 𝑋 and 𝑌 being an element of 𝑌.

So as you could see, I’ve written this definition down and I’ve used some set notations. I’ve got this giant ∈. What this means is an element of. So what this means in real terms is we’re gonna take the values from 𝑋 and 𝑌 and create ordered pairs of coordinates from them. So therefore, we’re gonna have a set of coordinates whose 𝑋-values are going to be eight and 𝑌-values are gonna be three or eight. So therefore, the Cartesian product of 𝑋 and 𝑌 is going to be eight, three and eight, eight because they are only possible pairs of products. Because we’ve got the eight from set 𝑋 with the three from set 𝑌 and then the eight from set 𝑋 with the eight from set 𝑌.

So now, we’re gonna move on to 𝑌 and 𝑍, so 𝑌 multiplied by 𝑍. So with the Cartesian product of 𝑌 and 𝑍, we’re gonna have the coordinates 𝑌, 𝑍, where 𝑌 is gonna be either three or eight and 𝑍 is gonna be either four, five, or nine. So therefore, our set of possible results is gonna be three, four; three, five; three, nine; eight, four; eight, five; and eight, nine. So now, what we’re going to do is find the union between these two sets. Union is this shape here we’ve seen as a set notation of a U. And the union of two sets means a value that is an element of either A or B. So an element of the first set or an element of the second set.

And if we look at it as a Venn diagram, we could see that if we had sets A and B, if we wanted the union of those two sets, it’s any value within those sets. So therefore, we can say that the union of 𝑋 multiplied by 𝑌 and 𝑌 multiplied by 𝑍 is gonna be all the ordered pairs in 𝑋 multiplied by 𝑌 or 𝑌 multiplied by 𝑍. So therefore, we can say that we’re gonna include all the values that we’ve got. So for the ordered pairs, we can say that the union of the Cartesian product of 𝑋 and 𝑌 and the Cartesian product of 𝑌 and 𝑍 is gonna give us the set of coordinates eight, three; eight, eight; three, four; three, five; three, nine; eight, four; eight, five; and eight, nine.