 All right, let's take a look at problem 110 and this is the following problem We're given a couple of different sets and we want to prove based only on the relevant definitions that the Cardinalities are whatever they happen to be so here the cardinality of B is equal to 2 We also want to prove this statement that 1 is less than 2 So as a general rule, it's important to remember that any question that asks you to prove something is Going to somehow be based on a definition And of course since we know the definition it is possible for us to do mathematics So if we want to prove that the cardinality of B is equal to 2 So I need to know the definition of 2 and we pull that up to is the cardinality of the set whose only proper subsets have Cardinality 1 or 0 so let's go ahead and take a look at that Definitions are very much like checklist if it fits the checklist if it satisfies all the requirements then the definition applies So let's check those proper subsets of B and see if they actually have cardinality 1 or 0 Now again since we know the definitions we can do mathematics the proper Subsets of B are the sets whose elements are in B, but they aren't the same as B the sets aren't the same So let's take a look at those proper subsets. So here's B I need a set whose elements are drawn from here So maybe that element is X I can't include this element Y because then that gives me something that's the same as B So that's a proper subset Cardinality is 1 because it's part of the definition Proper subsets have cardinality 1 or 0 because it's part of the definition I can just say that this has cardinality 1 without further comment Likewise another proper subset of B is just the element Y and again I can claim that cardinality is 1 and well I need to find all proper subsets So I need to see if there are any more proper subsets of B and let's see. So these are the Subsets that I can form by taking this element This element I can't take both elements So maybe I can take neither element that gives me the empty set and that is a subset with cardinality 0 So let's check our definitions the proper subsets of B have cardinality 1 or 0 So my definition applies and so the cardinality of B is equal to 2 It never hurts to include a summary sentence Now just as a matter of style, what do we need to prove cardinality B is equal to 2 all of this We have to include all of this as part of our proof So the highlighted portion is the proof nothing less than the highlighted portion is a complete proof Let's take a look at our second component our second thing. We're trying to prove we want to prove that 1 is less than 2 And so to prove 1 is less than 2 we need to refer back to the definitions So again our definition for less than suppose I have a 1 to 1 correspondence between a and a proper subset of B Then the cardinality of a is less than the cardinality of B So because the cardinality of a cardinality B are part of the definition I can simply claim their values a has cardinality 1 B has cardinality 2 And so I have my 1 less than 2 I just need to make sure that I fulfill the rest of the definition I need to find a 1 to 1 correspondence between a and a proper subset of B So let's see if we can set that correspondence up. So here's a and I want to find a 1 to 1 correspondence between a and a proper subset So maybe I'll show this element in a is going to be matched to Well, how about this element and So now I have a set here Everything in this set is an element of this set This set is not the same as this set So that says that this is a proper subset of B and again a summary never hurts Since there is a 1 to 1 correspondence between a and a proper subset of B Cardinality of a is less than cardinality of B Cardinality of a 1 I can replace it B Equals 2 so that tells me that 1 is less than 2 And again because this is a proof statement what we need to show is All of this again. We're showing the correspondence to a proper subset and we're including our summary statements Anything less than this will not be a complete proof You