 Let's continue our discussion of sets by looking at what happens when we try to find relationships between sets. And so this is a very common thing to do in mathematics, given an object that we can talk about. Let's consider the relationships between them. And so we might start off with several relationships we can have between sets. First off, two sets are equal if every element of the one set is also an element of the other set, and importantly, vice versa, every element of B has to also be an element of A. And if that happens, we write A is the sine equals B. Likewise, we can go farther. Maybe not every element is an element, but maybe we don't have the vice versa. B is a subset of A if every element of B is an element of A. And this is almost this definition, except we don't have the vice versa. We don't know or care whether A, every element of A is also an element of B. Now, if it happens to be that B is not equal to A, in other words, if we very definitely don't have the vice versa, we say that we have a proper subset. In either case, if B is a subset of A, we use the notation B subset sort of like a mutated less than or equal to sine A. We can also write it the other way. B is a subset of A. If I read right to left, it's the same sequence B subset A, but if I read from left to right, I might read this as A contains B. The distinction is important because how you speak influences how you think. Generally we like to write it this way. B is a subset of A. Every now and then it's convenient to write it the other way, but usually this will be our standard notation. For example, let's consider the following problem. I have a set S where what I'm looking at things which are U.S. states. I have a set W, things which are U.S. states east of the Mississippi River, and things which are cities in the northeastern part of the United States. So prove or disprove. Let's read this. W is a subset of S and is a subset of S. I'm not yet making a claim whether these are even true. I want to prove or disprove them. Now again, part of the reason for proof in mathematics is that it reminds us of stuff that we either have already learned or we should learn. So in this particular case to determine if W is a subset of S, well I can go back to see what the definition of subset is. So I'll review that definition of subset. B is a subset of A if everything in B is also an element of A. Well I want to determine if W is a subset of S, so I'll instantiate. I'll put W in the right place. I'll put S in the right place. And so I want to determine W is a subset of S if every element of W is also an element of S. Well let's see if that's true. Let's think about this. W consists of the states that are east of the Mississippi. S consists of all U.S. states. And here's my important observation. These are what W and S are. And let's try and relate our observation to our definition. Well everything in W is a U.S. state. So everything in W is going to also be an element of S. So W will be a subset of S and our proof is going to be the part in green. We have to include the entire observation. How about the other one? N subset of S. Well again we'll check to see if it fits the definition. And in this case we note that the elements of N are cities in the northeastern part of the United States. They are not states in any way, shape, or form. So the things that are in N are not in S. These are cities. These are states. And so again relating our observation to our definition gives us our conclusion. So we have the elements of N are cities. They are not states. So N is not a subset of S. And as before we can indicate the not relationship with the slash through the relation. And again the essential part of the proof is the portion in green, which must be included when we prove things.