NOTES ON FUNCTIONS DAMIEN PITMAN

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1 NOTES ON FUNCTIONS DAMIEN PITMAN Definition: We say R is a relation from A to B iff R A B where A and B are nonempty sets. The set A is the source for R and B is the target for R. For any source element a A and any target element b B, we say a is related to b by R or write arb iff (a, b) R. Definition: Given two relations R and S, we say R is equal to S iff xry xsy and the source of R is the source of S and the target of R is the target of S. Definition: Given a relation R from a A to B, the inverse of R, denoted R 1 is the relation from B to A satisfying arb br 1 a for every a A and b B. Definition: Given a relation R from A to B, for any set X A, we define the image of X under R as R(X) := {y B x X, xry}. We also define the inverse image of Y under R for any Y B as R 1 (Y). Definition: Given a relation R from A to B, the domain of R is R 1 (B) and the image of R is R(A). Comment: The domain and image of a relation are the values that actually appear as first and second coordinates in the relation. The source and target are nothing more than the names given to the sets chosen to contain the domain and image of the relation. Comment: In mathematics, the term domain is universally agreed upon to have the meaning defined here. The image of a set and inverse image of a set also have universally accepted meanings; but many authors use the term range for what we call the image of the relation, while others use the term range to refer to what we call the target. We have chosen to avoid using the term range because of its ambiguous meaning. The usual term used to refer to what we call the target is codomain. We have chosen to avoid using the term codomain because the term seems to put domain and codomain on equal footing, despite the fact that the domain is restricted and the codomain is not. The term 1

2 NOTES ON FUNCTIONS 2 source is less commonly used and there is no universally accepted term for this concept. Definition: An arrow diagram for a relation R from A to B consists of labeled source and target elements with arrows from domain elements to image elements. Comment: Notice that the image of a set X is the set of image elements with arrows coming from X. Since arb br 1 a for any a A and any b B, inverting a relation simply reverses the direction of the arrows and so the inverse image of a set Y can be thought of either as the image of Y under the inverse relation R 1 or as the set of source elements for R that relate to elements in Y. Symbolically, this is summarized by R 1 (Y) := {x A y Y, yr 1 x} = {x A y Y, xry}. Definition: A relation R from A to B is a function iff for each element of the source A relates to exactly one element of the target B. We write f : A B iff f is a function from A to B. Also, we write y = f (x) iff (x, y) f and f is a function, in which case we refer to y as the image of x and x as a preimage of y. Comment: In symbols, a relation R from A to B is a function iff R({a}) = 1 for each a A. For functions, the image of an element is always a single, unique element. However, there may be any number of preimages for a single element and R 1 will be a function iff R({b}) = 1 for each b B. Below, we define the terms one-to-one and onto so that for any function f : A B, being one-to-one is equivalent to f 1 ({b}) 1 for each b B and being onto is equivalent to f 1 ({b}) 1 for each b B. For any relation, the image of a set X is the collection of elements that elements from X relate to, which is the set of images of elements from A if the relation is a function. The image of a relation or function is the image of the source. In the case of a function, the source is the domain, so the image of a function is the image of its domain. Thus, the term image is used in three distinct, but related ways. The term preimage refers to any element in the inverse image of the set containing a given element, under a function. Comment: Many commonly encountered functions map elements from R to R and are defined by the rule that determines the image (output), f (x), for a given domain element (input) x. In this case, the domain is usually not stated, but rather understood to be the natural domain

3 NOTES ON FUNCTIONS 3 of the function, i.e., the largest subset R for which the rule f (x) makes sense. Comment: Two functions f and g are equal as functions when they are equal as relations. Thus, we write f = g exactly when f : A B and g : A B and f (x) = g(x) for all x A. Definition: A function f : A B is said to be one-to-one or injective iff for any a, x A, a = x implies f (a) = f (x). Comment: A function f : A B is one-to-one if and only if for any a, x A, f (a) = f (x) implies a = x. Theorem: A function f : A B is one-to-one if and only if f 1 ({b}) 1 for each b B. Definition: A function f : A B is said to be onto or surjective iff for all y B there exists x A, such that f (x) = y. Comment: A function f : A B is onto if and only if f (A) = B. Theorem: A function f : A B is onto if and only if f 1 ({b}) 1 for each b B. Theorem: (Pigeon-Hole Principle) If A and B are sets such that A > B, then there does not exist an injective function from A to B. Definition: Let f : A B and g : B C be functions. The composition of f with g, denoted g f : A C, is defined by g f (a) = g( f (a)) for each a A. Theorem: Let f : A B and g : B C be functions. (1) If f and g are injective, then so is g f. (2) If g f is injective, then f is injective. (3) If f and g are surjective, then so is g f. (4) If g f is surjective, then g is surjective. (5) If f and g are bijective, then so is g f. (6) If g f is bijective, then f is injective and g is surjective. Definition: Let A be a nonempty set, then the identity function on A, denoted id A, is defined by id A (a) = a for all a A Definition: A function f : A B is said to be invertible iff its inverse relation f 1 is a function from B to A. Theorem: For any function f : A B, the following conditions are equivalent. (1) f is bijective. (2) f is invertible. (3) f 1 is bijective. (4) f 1 f = id A and f f 1 = id B.

4 NOTES ON FUNCTIONS 4 Proof: The surjective property of f rules out the possibility that an element in the codomain/target B lacks a preimage in the domain A under f, which is equivalent to saying that every element of B is in the domain of the relation f 1, which means that f 1 satisfies the first necessary property for functions. The injective property of f rules out the possibility that two elements of the domain A map to one element of the target B, which is equivalent to the uniqueness of images under f 1, which means that f 1 satisfies the second necessary property for functions. Thus, if f is bijective, then f 1 is a function. Moreover, if f failed to be either injective or surjective, then f 1 would fail to be a function. From this we see that a function is bijective if and only if it is invertible. A moment s thought will also convince the reader that since ( f 1 ) 1 is f, it must be true that f 1 is a bijection. When considering the composition of a function with its inverse, a quick computation shows that for any a A, f 1 ( f (a)) = a and for any b B, f ( f 1 (b)) = b, which verifies that last property of bijective/invertible functions. All that remains to prove our theorem is that the last property implies any one of the others. But f and f 1 must be bijections by the previous theorem because their compositions in both orders are identity functions, which are bijections. Proof: (Alternate for (1) (2)) Arguing simply on cardinalities, we see that f is onto if and only if f 1 ({b}) 1 for each b B and f is one-to-one if and only if f 1 ({b}) 1 for each b B. Thus, f is bijective if and only if f 1 ({b}) = 1 for each b B, which is equivalent to f 1 being a function. Definition: For any two nonempty sets A and B, there is a collection of functions from A to B. We denote this set by B A = { f f : A B}. Proposition: For any two nonempty sets A and B, the cardinality of the set of functions from A to B is B A = B A. Comment: When B = {0, 1} it is customary to denote B A as 2 A.

5 NOTES ON FUNCTIONS 5 Examples Example: Find all functions from {a, b} to {1, 2, 3} and determine which are injective. Example: Find all functions from {a, b, c} to {1, 2} and determine which are surjective. Example: Find all bijections from {a, b, c} to {1, 2, 3}. Example: Determine why there were no injective functions in the first example and no surjective functions in the second. Example: Find the inverse function for one of the functions in the last example and discuss why there are no invertible functions in the first two examples. Example: Find both compositions of f (x) = x 3 and g(x) = x 4. Example: Explain why compositions of the arrow diagrams above do not exist. Then create some compositions from {a, b} to {α, β, γ} to {0, 1}. Find an example where the composition is bijective, but one of the composite functions is injective and the other is not surjective. Example: Let f (x) = 5x 2. Show that f : R R is a bijection and find its inverse. Example: Consider the set {(m, n) m = 3n, n N}. List the elements of this set and graph it. Then, thinking of it as a function, find the domain and image. Describe its inverse function. Example: Consider the equation y = x 2. What ordered pairs solve this equation? Graph the solution set. That set of ordered pairs is the function. Show that f : R R defined by f (x) = x 2 is neither onto, nor one-to-one. Example: Consider the rule f (x) = (x 3) What is the natural domain of this function? What is the image of the function from its natural domain? Show that f : R R is neither onto, nor one-to-one. Example: Consider the rule g(x) = x What is the natural domain of this function? What is the image of the function from its natural domain? Show that g is a bijection, find g 1 and directly show that g g 1 and g 1 g are identity functions, being careful to define the sets on which they are identities. Example: What is the inverse rule of f (x) = e x? Carefully define this inverse pair of functions.

6 NOTES ON FUNCTIONS 6 Example: Consider the rule f (x) = x 3. What ordered pairs does this equation produce? Graph the function. What is the natural domain of this function? What is the image of the function from its natural domain? Show that f : R R is not onto, but is one-to-one. Show that f is a bijection from its natural domain to its image from that domain; then describe the inverse function and directly show that f f 1 and f 1 f are identity functions, being careful to define the sets on which they are identities.. Example: Consider the rule: map each integer to its remainder when divided by 5 (as in in the division algorithm). Does this rule determine a function? If so, what is the domain and what is the image? Is it a bijection? Is it invertible? What is the inverse image of 0? of 1? Definition: A function f on equivalence classes is said to be well-defined when the choice of representative for an equivalence class in the domain does not affect the image under f. Example: Let f : Z 3 Z 3, f ([x]) = [2x + 1]. Prove that this is a well-defined bijection. Example: Show that f : Z 3 Z 3 defined by f ([x]) = [x 2 ] is well defined, but is not a bijection. Example: Let f : Z 3 Z 3 be defined by f = g h where h([x]) = [2x] and g([x]) = [x + 1]. Is this well-defined?