Lecture 20 21 127 Concepts of Math 10 15 2012 Administrivia Homework comments If you collaborate with like 10 people at once and you all come up with the same mistake you re gonna have a bad time Limit your collaborations to no more than 3 or 4 people at a time Feel free to talk about the problems with several different people but don t get a huge group together in one room to work together Groupthink happens people Pay careful attention to the definitions I give you in the notes as well as how you use them We ve seen several mistakes that amount to just misunderstanding the definition which leads to a misinterpretation of the problem statement or some important part of the solution Just talk to us if you want to reinforce a concept Have an example and a non example in mind for every definition we discuss There is nothing about the quantifier that says the objects you are quantifying need be distinct You might be thinking that because they are different letters but that is just not true The following statement is False x y N x y y x because when x y the statement fails but the following statement is True x y N x 6 y x y y x In general the quantifier says For any possible choices of these objects even if some of them might be the same If you want distinctness as an important property in your statement you have to build it in like I did in the example above Images and Pre images Definitions Definition Let A B be sets and let f A B be a function Let X A The image of X under the function f is the set of all outputs of f considering all the elements of X as inputs More rigorously the image of X under f is written and defined as Imf X b B a X f a b That is Imf X is the set of all elements b of the codomain for which we can find an input a A such that f a b In more loose terminology we might say that the image contains all of the elements of the codomain that are hit by the function from the elements of the domain that belong to X An equivalent way of writing this set is Imf X f a a X Definition Let A B be sets and let f A B be a function Let Z B The pre image of Z under the function f is the set of all inputs of f that land in Z More rigorously the pre image of Z under f is written and defined as PreImf Z a A f a Z 1 Examples Example Define a function g A B by setting A a b c d e and B 1 2 3 4 5 6 and g a 2 b 3 c 3 d 1 e 6 Define X1 a b c and X2 a c e and X3 c d e Define Z1 1 2 3 and Z2 2 3 4 and Z3 4 5 6 Let s identify the following images and pre images and explain Notice that we have dropped the subscript Img and PreImg for convenience throughout this example we are only referring to g so we ll just leave that off 1 Im a 2 since g a 2 2 Im b c 3 since g b g c 3 3 Im X1 2 3 4 Im X2 2 3 6 5 Im X3 1 3 6 6 PreIm 1 d since g d 1 and no other x A satisfies g x 1 7 PreIm 4 since no x A satisfies g x 4 8 PreIm Z1 a b c d since g a 2 g b g c 3 g d 1 but no other x A satisfies g x C1 9 PreIm Z2 a b c since g a 2 and g b g c 3 but no other x A satisfies g x Z2 10 PreIm Z3 e since g e 6 but no other x A satisfies g x Z3 11 PreIm 5 because g x 6 5 for every x A On the homework you are supposed to investigate and prove the claims you come up with what happens when we find the image of an intersection or a union of two subsets of the domain Example Consider the range of temperatures between when water freezes and when it boils Specifically define the set C x R 0 x 100 and define the function F C R by c C F c 9 c 32 5 What is ImF C What does it represent To approach questions like these we must a identify a claim for what ImF C is and then b prove that the set we identified is actually equal to ImF C in the sense of sets 2 Proofs Lemma Let f A B be a function Let Y Z B Then PreImf Y Z PreImf Y PreImf Z Proof First we show Let x PreImf Y Z This means f x Y Z and so f x Y and f x Z Since f x Y this means x PreImf Y Since f x Z this means x PreImf Z Since x PreImf Y and x PreImf Z this means x PreImf Y PreImf Z Second we show Let w PreImf Y PreImf Z This means w PreImf Y and w PreImf Z Since w PreImf Y this means f w Y Since w PreImf Z this means f w Z Since f w Y and f w Z this means f w Y Z This means w PreImf Y Z Important lesson Just follow the formal definitions Write down what we know and what we want Perhaps have an example in mind on your scratch paper to make sure what you say makes sense Lemma Let f A B be a function Let Y B Then Imf PreImf Y Y and equality need not hold Proof Let b Imf PreImf Y This means a PreImf Y f a b Let such an a be given Since a PreImf Y this means f a Y Since b f a this means b Y To show equality need not hold consider the following counterexample Define A 1 2 3 and B Define the function f A B by f 1 2 3 Define Y Notice that PreImf Y 2 3 since f 2 f 3 Y but f 1 Y Then notice that Imf PreImf Y Imf 2 3 Y Important lesson Use a schematic diagram to help yourself come up with a counterexample It doesn t have to be an interesting one just one that works Remember that a function is not necessarily a rule for assigning values you can define a function by just telling me the related pairs Properties of Functions Surjectivity and Injectivity Definitions Motivation By definition for a function f A B we have Imf A B We might wonder then when we have …