 All right, so here's a little guidance for problem 215. Now, again, don't watch this video. Problem solving is to develop your own method of approach, and as soon as you watch this video, you will not be able to solve the problem. Again, remember, applying a solution given to you by somebody else is not problem solving and is not mathematics. But let's say we're absolutely stuck on this and so we want to try and solve this problem and this is to prove A plus 0 is equal to A. And again, because it's a prove question, we have to go back to our definition of what addition is. So the addition is going to be defined as follows. If I have a set with a cardinality, another set with a cardinality, the two sets have nothing in common, then addition defines the sum of the cardinalities to be the cardinality of the union of the two sets. So what I need is I need a set with cardinality A. Now, because this is a generic question, A is anything, I have to use a generic set A and I know that any set has a cardinality. So I can take any set I want to and let that be my set A and it's important because my set is generic, I can't say anything about what A is. On the other hand, because I'm adding 0, because I'm adding 0, I need a set whose cardinality is 0. Well, by definition, the empty set has cardinality 0. So I'm going to let my second set be the empty set and now I can apply the definition. So A plus 0 is going to be the cardinality of the union of the two sets. So A plus 0 is the cardinality of the union of the set A with the empty set. Well, the set A together with the empty set is just going to be the set A. Remember, if it's part of the definition, we don't really have to comment on it other than to just claim a statement and make sure that that statement is actually true. A union of the empty set is going to be A because remember, this is the set of everything that's in A together with anything that's in the empty set. Well, the empty set doesn't have anything in it. So when I put these two things together, I just get the original set. So A plus 0 is the same as the cardinality of A. Again, the equality says any time you see the one thing, you can immediately replace it with the other. So here I have A union of the empty set. I can replace it with just A. And so A plus 0 is the same as the cardinality of A. But I already know what that is. The cardinality of A is A. And again, the equals says I can replace the one thing with the other. So instead of cardinality of A here over on the right-hand side, I'll replace it with A. And there's my statement. The last thing that we write down is what we've actually proven and that's what we actually want to do.