If 𝑋 is equal to the intersection between the set containing the elements seven and one and another set containing the elements seven and three, then nine what 𝑋?
The gap in this problem, which we read as what when we read the question, could be completed by using one of two symbols. Either nine is an element of 𝑋 or nine is not an element of 𝑋. Let’s go through the problem slowly and try to understand it because it’s full of symbols and not a lot of words. And, hopefully, by doing this, we can work out which of the two symbols to use.
To begin with, we’re given a string of numbers and symbols and told that they’re equal to 𝑋. The statement contains two sets. The first set contains the numbers seven and one, and the second set contains the numbers seven and three. This upside-down U shape in between both sets represents the intersection between them. That’s why we read the question as if 𝑋 is equal to the intersection between a set containing the numbers seven and one and another set containing the numbers seven and three. These questions all about the part where these two sets overlap or intersect.
The best way to think about an intersection is to consider a Venn diagram. Our first set contains the number seven and one, and our second set contains the number seven and three. The intersection between the sets is the part which overlaps here. And because the number seven is in both sets, we can write this in the central section. Now we’ve labelled our Venn diagram correctly. As we’ve mentioned already, we’re told in the question that 𝑋 is found at the intersection between both sets. In other words, it’s found in this overlapping part of our Venn diagram. The common element between both sets, the part where they intersect, is the number seven. So 𝑋 is a set that just contains the number seven.
Now that we know that 𝑋 is the same as seven, we can go back to the problem and work out which symbol to use. We’re told nine something 𝑋. Now we know what 𝑋 represents. So we can rub it out and replace it for a set that includes the element seven. Remember the two symbols we said we could choose from was this one, which means “is an element of,” and this one, which means “is not an element of.” So is the number nine an element of a set which just contains the number seven or is the number nine not an element of a set that contains just the number seven?
We can see that we need to complete the statement using the second symbol. If 𝑋 is a set that only contains the number seven, then nine is not an element of it. We recognize that 𝑋 was found at the intersection between a set containing the numbers seven and one and another set containing the numbers seven and three. And we decided that the element that was shared between both sets was the number seven. So 𝑋 must equal seven.
If 𝑋 is a set containing the number seven on its own, then nine is not an element of it. So we use the symbol is not an element of to complete the statement.