Use the symbol for is a subset of, is not a subset of, is an element of, or is not an element of to fill in the gap. The Venn diagram shows 𝑍 what 𝑌.
In this problem, we need to think about two sets, 𝑍, and we can see that the part of our Venn diagram that’s labelled 𝑍 is the square. So everything within the square is part of set 𝑍. So we can say that set 𝑍 is made up of the elements three, one, four, five, six and nine. That’s everything within the square. And then, the second set that we need to think about is set 𝑌. And, again, we can read the Venn diagram to see what elements set 𝑌 contains. Set 𝑌 is the orange circle. And this contains three elements: five, six and nine. And the sentence we need to complete compares set 𝑍 to set 𝑌. The Venn diagram shows 𝑍 what 𝑌.
And we have four possible symbols to complete this sentence with. Let’s spend a moment to remind ourselves what these symbols mean. We can split them into two groups. Our first two symbols are the ones that we use to show whether a set is or is not a subset of another set. When we use the word subset, we mean that that set is completely within another set. Our second pair of symbols are the ones that we use to describe whether something is or is not an element of a set. In our example Venn diagram, we can see that the number three is an element of set 𝑍. It’s one of the numbers that’s contained within it.
Let’s start by thinking about whether our missing symbol is one of these two symbols. Well, if we look closely at our statement, it’s talking about set 𝑍 and set 𝑌. These are two sets. These are not individual numbers or individual elements. So we can’t say that set 𝑍 is an element of set 𝑌 or is not an element of set 𝑌. It’s a set; it’s not an element. So it would be wrong to use either of these two symbols to complete our sentence. And so, we’re left with one of our first two symbols. Is 𝑍 a subset of 𝑌 or not a subset of 𝑌?
Well, if we look at our Venn diagram, we can see that one of the sets is a subset of the other. But the letter should be the other way around. We can see set 𝑌 is inside set 𝑍. And we can show this using our notation too. The numbers five, six, nine are also in set 𝑍. So we could definitely say that 𝑌 is a subset of 𝑍. But our letters are the other way around. On our Venn diagram, set 𝑍 is the largest set. It can’t be a subset of 𝑌. So the correct symbol to use is the one that represents is not a subset of. We know that set 𝑍 contains six elements: three, one, four, five, six, and nine. But set 𝑌 only contains three elements: five, six, and nine. So set 𝑍 is not a subset of set 𝑌.