 The 19th century American mathematician Benjamin Perce once said, mathematics is the science of necessary consequences. What he meant is that in mathematics you begin with a few ideas and follow the logical consequences of those ideas no matter where they lead. So let's consider an important concept and see where it takes us. We say that there is a one-to-one correspondence between two sets A and B if every element of A can be matched to a unique element of B and also every element of B can be matched to a unique element of A. For example, let's try to prove or maybe disprove. There is a one-to-one correspondence between the sets A equal spade black square five and B the set of things where thing is a letter in the word cat. Since A is given in list notation it might help to put B in list notation as well and so B is and we want to match every element of A with a unique element in B. So let's write down our two sets and we'll exercise our creative powers to the maximum and we might match spade with how about C and we'll represent that by drawing an arrow from spade to C and then black square we could match that to A and again we'll draw an arrow from black square to A and five could be matched to T and again we'll draw an arrow from five to T. Now this matches every element of A with a unique element of B. We also want to go in the other direction. We want to match every element of B with a unique element of A. Now there's many ways we could do this. For example we could match C with black square A with five and T with spade but we could also just follow the original correspondence backwards and so match C back to spade A back to black square and T back to five and to avoid too much clutter on our diagram we can replace this up and down arrow with a single double-headed arrow and so every element of A is matched to a unique element in B and every element of B is matched with a unique element in A and so this is a possible one-to-one correspondence. Now something we might try to describe about a set is the number of elements it contains. However, number is so fundamentally a concept that it's very difficult to define and instead we'll use this idea of cardinality. Two sets A and B have the same cardinality if there is a one-to-one correspondence between them and we write it this way. Now you might recognize these vertical bars from another context and may read them as absolute value and the important thing to remember is there are only so many symbols. Sometimes we have to reuse them in a totally different context and remember how you speak influences how you think. You want to be very careful not to read this as absolute value of A but rather cardinality of A. Now you might wonder how you can tell whether this means cardinality or absolute value and here's the important thing. Context is everything. If they show up in the context of questions about cardinality or about sets then it refers to cardinality. Now this might seem to be a very roundabout way of talking about a very fundamental concept but the advantage to using a one-to-one correspondence is we know if a one-to-one correspondence exists because if we find one it exists. So for example let's go back to our two sets and let's try to prove or maybe disprove that these two sets have the same cardinality. So remember definitions are the whole of mathematics all else is commentary. Our definition says that two sets have the same cardinality if there is a one-to-one correspondence between them. So to show that these sets have the same cardinality we have to show that there is a one-to-one correspondence between them. Well we already found one and since there is a one-to-one correspondence the two sets have the same cardinality. Or let's take a different problem. Suppose S is the set of things where what we're talking about are states in the United States in the year 2050 and C be the set of things that are capitals of a US state in the year 2050. And let's prove or disprove the cardinality of S and C are the same. So to do this all we need to do is find a one-to-one correspondence between every state in the US in the year 2050 and every capital of a US state in the year 2050. So let's go to the year 2050. Oh we can't do that. Well we'll just wait around until the year 2050 to solve this problem. Okay maybe we want to get the answer before then. So we seem to have a problem. But remember definitions are the whole of mathematics all else is commentary and we have our definition for cardinality. So as an alternative we can describe how we'll make a correspondence then make an argument that the correspondence we make is a one-to-one correspondence. We need some way of matching every state in S with a city in C. Now while we could match a state with a random capital it might make sense to match the state with its own capital. And that capital is going to be unique. A state would only have one capital. And so we claim we can match every state in the US in the year 2050 with its unique capital city. We also have to go the other way. Every city that is the capital of a US state needs to be matched to a unique state. And again while we could match it to any state we wanted we should probably match it to the state it's the capital of. And there's no other state that it would be the capital of so this matching would be to a unique state. And so we can match every city that is the capital of the US state in 2050 to the unique state it's the capital of. And so every element of S will correspond to a unique element of C that's this matching of states to their own capitals. And every element of C will correspond to a unique element of S that's this matching of the capital cities to the states that they're capital of. And we've met the requirements and so the cardinality of S and C is the same.