Video Transcript
Given that 𝐴 is a set containing the elements three, six, seven, and nine; 𝐵 is a set containing the elements three, five, six, and nine; and 𝐶 is a set containing the elements five, six, and nine, write the set of common elements in 𝐶 and 𝐵. Then write the set of elements common to 𝐴 and 𝐵 and 𝐶.
This question gives us information about three sets. We’re calling them 𝐴, 𝐵, and 𝐶. And each set contains a number of numbers. Set 𝐴 and 𝐵 both contain four numbers. Set 𝐴 contains the numbers three, six, seven, and nine. Then in set 𝐵, we have the numbers three, five, six, and nine. We’ll see that set 𝐶 is slightly smaller. It contains three numbers, five, six, and nine.
In the first part of the problem, we’re asked to write the set of common elements in 𝐶 and 𝐵. When we talk about common elements, we’re talking about the elements in a number of sets that are in all of the sets; they have them in common. A really useful way of illustrating sets is by drawing a Venn diagram. And we know that the common elements can be found in the part that overlaps. These are elements common to both sets.
So, let’s draw some Venn diagrams to help us solve this problem. As we’ve said already, the first part of the problem asked us to find the set of common elements in 𝐶 and 𝐵. Let’s let this orange circle represent set 𝐶, and we’ll let the pink circle represent set 𝐵. Now, we know 𝐵 contains the elements three, five, six, and nine. But if we look at the elements in set 𝐶, they are some of the numbers that we’ve already mentioned for set 𝐵; they’re in both sets.
So, we need to write the numbers five, six, and nine in the overlapping part of our Venn diagram. These are the common elements in set 𝐶 and set 𝐵. And so, we can write the set of common elements in sets 𝐶 and 𝐵 as a set that contains the elements five, six, and nine.
But there’s a second part to our problem. We’re asked to write the set of elements common to 𝐴 and 𝐵 and 𝐶. This means we’re introducing set 𝐴 too. Our Venn diagram is going to look a little bit different. Let’s include a green circle to represent set 𝐴. Now, set 𝐴 contains the elements three, six, seven, and nine. Let’s think about each of these numbers in turn.
The number three is also in set 𝐵, but not in set 𝐶. So, this is a common element between sets 𝐴 and 𝐵. We’ll write the number three here. The next number, six, is in set 𝐵 and also in set 𝐶; it’s a common element between 𝐴, 𝐵, and 𝐶. And so, we must write the number six in the centre section where all three circles in the Venn diagram overlap. Then we have the number seven. The number seven isn’t in set 𝐵 or set 𝐶. So, this is just in set 𝐴. And finally, the number nine. And as we’ve discussed already, the number nine is in set 𝐵 and set 𝐶. So, it’s another one that belongs in this centre section.
So, so far, we’ve put the numbers three, six, nine, and seven onto our Venn diagram. The only number that we’re now missing is the number five. And again, as we’ve talked about already, this is in sets 𝐵 and 𝐶. So, we write the number five in this overlapping part here. Now, we can solve the problem. The set of elements common to 𝐴 and 𝐵 and 𝐶 are going to be those numbers that are in all three circles. And that’s the overlapping part right in the middle of our Venn diagram. There are only two elements there. And they are the elements six and nine.
We modelled the Venn diagram to help us solve the problem. The set of common elements in sets 𝐶 and 𝐵 is a set containing the numbers five, six, and nine. And then, the set of elements common to 𝐴 and 𝐵 and 𝐶 is a set containing the numbers six and nine.
