 So welcome back to our last video about inverses. So far we've seen that we can represent functions as sets of ordered pairs. And that if we take a function represented in this way and flip the coordinates on each pair, we get this new set called the inverse of F, which we denote F with a little negative one. And remember that negative one is not an exponent that indicates fractions are going to take place. And sometimes, but not always, F inverse is actually a new function in its own right, and this happens precisely when both, when F is both injective and surjective. So in this video, we're going to examine this situation closely and ask about how we work with inverses when they turn out to be functions in their own right. How do inverse functions work and how do we prove results that involve them? So for the duration of this video, let's assume that F going from A to B is a bijective function. That means that not only does F inverse exist as a set of pairs, but that F inverse is a function itself. Now let's think about whether we have all the ingredients we need for a successful function. In particular, what's the domain? What's the co-domain and what is the process by which F inverse works? We don't need to worry about the other two ingredients, whether F inverse maps each point to the domain to something and whether F inverse splits inputs or not because those properties are taken care of when we assume that F, the original function, is bijective. We discussed that in the last video. So what are the domain and co-domain of F inverse? This is easy to answer if you remember the pairs that are in F inverse. For the original function F, we represented that as a list of pairs. F equals the set of all pairs AB and A cross B such that B equals F of A. So the first coordinate comes from the domain A and the second coordinate comes from the co-domain B. With F inverse, the orders are reversed here. So the first coordinate would now come from B and the second coordinate comes from A. So that means that the domain of F inverse is B and the co-domain of F inverse is A, which of course is precisely the opposite of the domain and co-domain of the original function F. So that's easy enough. Let's think about how the process for F inverse works. We define F inverse formally to be the set F inverse is the set of all ordered pairs B comma A and B cross A such that F of A equals B. Now, what does this mean? If F inverse is a function from B to A, which it is, then what is F inverse of B for a point B in the set B? Which remember is the new domain here. So the definition tells us that F inverse of little B equals A, if and only if F of A equals B. What this means for us is that to determine what F inverse of B is, we know it's going to be some point in A. So to determine the value of that point, we ask, what A in the set A gives us B when we load it into F? In other words, F of what equals B? The answer to that what question is the value of F inverse of B? That's tricky, so let's look at an example. Let F be a function from Z4 to Z4 and let it be defined by the table that you see here. As a set of ordered pairs, F would be the set F equals the set 0, 1, 1, 2, 2, 0, and 3, 3. And we can see directly that F is a bijection. So F inverse is going to exist and be a function in its own right. So what is the value of F inverse of 2? Well, what the previous slide was telling us is that whatever F inverse of 2 is, let's just call it U for the time being, F inverse of 2 equals U if and only if F of U equals 2. So I need to figure out what I put into F to give me 2. And I can read that straight off the table or from the list of pairs that that value is 1. F of 1 equals 2, and so therefore F inverse of 2 equals 1. Likewise, F inverse of 0 equals 2 because F of 2 equals 0. F inverse of 1 equals 0 because F of 0 equals 1. And F inverse of 3 equals 3 because F of 3 equals 3. Now this makes sense because when you put the, when you form the actual set of ordered pairs for the inverse, which remember is just a matter of reversing the orders of the pairs you have in F, you get F inverse equals the set 1, 0, 2, 1, 0, 2, 3, 3 just by swapping the coordinates. And you can check that this is a function 2 as is predicted. And all the outputs of this function are what we said they would be. F inverse of 1 equals 0 for instance. So let's see how well you're understanding this fairly tricky idea with a concept check. Let F be a function that goes from the real numbers into the open interval from 0 to infinity. Remember, this thing here is the set of all points from 0 to infinity, not including zeros, they're all positive real numbers. And we're going to define this function F by F of t equals 2 to the t power. Now you can check, and it's a good exercise, that F actually is a bijection from F into this open interval. And so F inverse is a function that goes the opposite direction from the open interval 0 to infinity to r. What is the value of F inverse of 32? Here's a list of points here, a list of values, and pause the video and make your selection. So the answer here is F inverse of 32 equals 5. Now, why is this? We're just going to use the rule for calculating F inverse. Whatever F inverse of 32 is, let's call it u for the time being. Then F inverse of 32 equals u means F of u equals 32. So I'm looking for a number u, such that when I put it into u, that would give me 2 to the u power, and that would equal 32. So in other words, I want to solve this equation. I can solve this equation using logs, or I can just use my knowledge of powers of 2 and realize that 2 to the fifth power is 32. So that means F inverse of 32 is 5, because F of 5 is 32. Now this illustrates an important point about inverses. We use them all the time to solve equations. And especially, this is where our definition of logarithms come from. The logarithm base two is defined to be the inverse function for F of x equals 2 to the x, and similarly for other bases of logarithms. So now that we have a handle on the rule for how F inverse works, if it's a function, let's use that rule in a proof involving inverse functions. Now here's a piece of a theorem, it's theorem 6.25 in your book, that will just detach from the main proof and prove as a standalone result. It says let F going from A to B be a function, that's actually a bijection. Then the inverse function F inverse, which maps B into A is surjective. In fact, F inverse turns out to be a full blown bijection going in the opposite direction. You can read the whole thing in your book, but let's just prove this part. So we already know that F inverse is a function. What we need to prove is that F inverse mapping B into A is surjective. So to do this, we're going to do it the way we always prove surjections, which is we choose a point in the co-domain, that's A this time. And construct a point in the domain, which is B this time. Let's call that point little b such that F inverse of B equals A. Now, how do we know such a point exists? Well, I've chosen A in A, that's the first step of my proof, to pick a point out of the co-domain. That said, capital A is the co-domain of F inverse, but it's also the domain of F. Now, since F is a function, F must map A to something in B. Let's say F of A equals Y in B. But now the proof is basically automatic. Since F of A equals Y, the definition of inverses tells us that F inverse of Y equals A. And then we're done because we started with A and found a point in B, namely little y, that maps to it. Well gosh, that was so easy, we might as well prove that F inverse is injective too. So to prove that F inverse is injective, let's suppose that we have two different points in the domain of F inverse, and let's prove they don't collide. Now the domain of F inverse is B, so let's call those two points B1 and B2. And again, assume that B1 is not equal to B2. I would like to prove that F inverse of B1 is not equal to F inverse of B2. Now for a contradiction, let's suppose that those two outputs were equal, so that F inverse of B1 does equal F inverse of B2. We'll try to do some math and come up with a contradiction here. Now these two points are equal, so let's designate their mutual value with X. Now let's note that X belongs to A. Since F inverse of B1 equals X, we use the definition of inverse, is to say that F of X equals B1. But since F inverse of B2 also equals X, it means that F of X equals B2. But this is a contradiction because I assumed F is a function. But here I have F of X, the same X, equaling two different things. F is splitting the input of X into both B1 and B2. And that's not allowed because B1 is not equal to B2 by assumption and F is a function by assumption. So this is a contradiction. So it cannot be the case that F inverse of B1 equals F inverse of B2. Hence they must be unequal and that proves that F inverse is injective. So put those two things together and it proves that if F is bijective, not only does F inverse exist and is a function in its own right, but it's a bijective function too. And we prove all that using just the straight definition of how to work with inverse functions. So that is a complete beginning look at the concept of inverses. Thanks for watching.