Lecture 23 21 127 Concepts of Math 10 24 2012 Administrivia Extra office hours this week Wednesday 2 30 3 30 and 4 30 5 30 Thursday 12 00 2 00 and 4 00 6 00 Inverse Functions In class on Monday we proved that f is a bijection f is invertible and furthermore that when f 1 1 exists it is also a bijection and f 1 f Example Let s practice with one example of finding the inversion of a function whose domain codomain are sets of ordered pairs Example Consider the function f R R R R given by x y R R f x y 2x y 2y x Let s try to find the inverse of f The idea is to take an arbitrary u v R R the codomain and find the corresponding x y R R the domain that would output that u v This amounts to setting f x y u v and solving for x and y This yields a system of two equations in two unknowns 2x y u 2y x v We can tackle this using algebraic methods or Gaussian elimination in general if you re familiar with that although that s really overkill when we only have two equations Subtracting the second from two times the first cancels the y and lets us solve for x 2 2x y 2y x 2u v 4x 2y 2y x 2u v 5x 2u v x 2u v 5 Adding twice the second equation to the first cancels the x and lets us solve for y 2x y 2 2y x u 2v 2x y 4y 2x u 2v 5y u 2v y u 2v 5 This scratch work now tells us how we want to define the inverse of f We can start the proof Proof Let s define F by F R R R R u v R R F u v 2u v u 2v 5 5 We need to argue that this is a well defined function and it clearly is because numbers for any u v and there s only one output Next we need to show that F f IdR R and f F IdR R as well Let s show one direction and leave you to figure out the other one 1 2u v 5 and u 2v 5 are real Let x y R R the domain of f Observe that F f x y F f x y F 2x y 2y x 2 2x y 2y x 2x y 2 2y x 5 5 4x 2y 2y x 2x y 4y 2x 5 5 5x 5y 5 5 x y This shows F f IdR R We will leave the proof taht f F IdR R as an exercise for the reader Note There is a more general Theorem which says that a function of this form f x y ax by bx ay is always a bijection provided that a and b are not both 0 which is stated by ensuring a2 b2 6 0 Notice that a 2 and b 1 here so we see a2 b2 5 and that is what appears in the denominator which is why we want that term to be nonzero Try to prove this more general result by finding the inverse of f Result Composing Bijections This next result is very helpful when combining bijections You are proving on the Prep Questions that g f is a bijection whenever f and g are both bijections so it is natural to wonder what the inverse of this composition will be since it is a bijection so it has an inverse This Theorem tells us what it is This will be very useful in the next section when we discuss Cardinality because we will be working with lots of bijections between different sets and this result tells us how to combine them appropriately Theorem Let f A B and g B C be bijective functions Prove that g f 1 f 1 g 1 Proof Notice that g f A C Let x A be arbitrary and fixed Observe that f 1 g 1 g f x f 1 g 1 g f x f 1 IdB f x f 1 IdB f x f 1 f x x Now let y C be arbitrary and fixed Observe that g f f 1 g 1 y g f f 1 g 1 y g IdA g 1 y g IdA g 1 y g g 1 y y Notice that we have applied the associativity of composition several times Good thing we proved that This will be helpful when we have bijections between three sets and we want to undo them This will help us come up with explicit bijections between two sets A and C where we couldn t quite see how to do it and needed to use an intermediary set B to do so Furthermore the definitions of Cardinality see below have a specific order to them to show a set is countable we need a bijection from S to N and not the other way around This Theorem we just proved tells us how to switch the order of a bijection defined as a composition 2 Cardinality Why do we care about bijections They allow us to compare the sizes of sets Pretend I didn t know how to count How could I show that there are just as many pieces of chalk on this table as there are erasers Exactly I could pair them off one by one In the language of sets I am identifying a bijection between the set of chalk pieces and the set of erasers In the land of Cardinality the Bijection is King We will use this language of functions and bijections to talk about cardinality the study of the sizes of sets both finite and infinite We need to use this terminology too because there are some surprising and unintuitive results and using these formal definitions and concepts allows us to be rigorous and precise about these results They might blow our minds a little bit but having them rooted in things we ve already seen and proven lets us belive those results mathematically speaking Definitions and notation Let S be any set We say S is finite if n N 0 such that there exists a bijection f S n Note The empty set S is finite since 0 We say S is infinite if S is not finite that is if n N 0 every function f S n fails to be a bijection We say S is countably infinite or just countable if there exists a bijection f S N We say S is uncountably infinite or just uncountable if every function f S N fails to be a bijection We use S to indicate the cardinality of S When S is finite so there is some n N 0 and a bijection f S n we write S n to mean that S has n elements We say n is the size of S Note It s interesting to think …