Page 1 of 9 Discrete Mathematics Functions Definition: A function 푓 from a set 퐴 to a set 퐵, denoted 푓: 퐴 → 퐵 is a well-defined rule that assigns each element of 퐴 to exactly one element of 퐵. We write 푓(푎) = 푏 if 푏 is the unique element of 퐵 assigned by the function f to the element 푎 of 퐴. Example: Recall: Let 퐴 and 퐵 be sets. The Cartesian product of 퐴 and 퐵, denoted by 퐴 × 퐵, is the set of all ordered pairs (푎, 푏), where 푎 ∈ 퐴 and 푏 ∈ 퐵. We can define a function 푓: 퐴 → 퐵 as a subset of the Cartesian product 퐴 × 퐵. © 2020, I. Perepelitsa Page 2 of 9 Given a function f : A → B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f. a is called the preimage of b. The range of f is the set of all images of points in A under f. We denote it by f(A). Two functions are equal when they have the same domain, the same codomain and map each element of the domain to the same element of the codomain. Example: Let 푓: 퐴 → 퐵, where 퐴 = {푎, 푏, 푐, 푑}, 퐵 = {푥, 푦, 푧}, and 푓(푎) = 푧, 푓(푏) = 푦, 푓(푐) = 푧, 푓(푑) = 푧. a. 푓(푎) =? b. The image of 푑 is? c. The domain of 푓 is? d. The codomain of 푓 is? e. The preimage of 푦 is? f. 푓(퐴) = ? g. The preimage(s) of 푧 is (are)? h. The range of 푓 is? © 2020, I. Perepelitsa Page 3 of 9 Definition: A function f is said to be one-to-one, or injective, if and only if 푓(푎) = 푓(푏) implies that 푎 = 푏 for all 푎 and 푏 in the domain of 푓. A function is said to be an injection if it is one-to-one. Example: Definition: A function 푓 from 퐴 to 퐵 is called onto or surjective, if and only if for every element 푏 ∈ 퐵 there is an element 푎 ∈ 퐴 with 푓(푎) = 푏. A function f is called a surjection if it is onto. Example: Definition: A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto (surjective and injective). Example: © 2020, I. Perepelitsa Page 4 of 9 Suppose that 푓: 퐴 → 퐵. To show that 풇 is injective: Show that if 푓(푥) = 푓(푦) for arbitrary 푥, 푦 ∈ 퐴, then 푥 = 푦. To show that 풇 is not injective: Find particular elements 푥, 푦 ∈ 퐴 such that 푥 ≠ 푦 and 푓(푥) = 푓(푦). To show that 풇 is surjective: Consider an arbitrary element 푦 ∈ 퐵 and find an element 푥 ∈ 퐴 such that 푓(푥) = 푦. To show that 풇 is not surjective: Find a particular 푦 ∈ 퐵 such that 푓(푥) ≠ 푦 for all 푥 ∈ 퐴. Example: Let 푓 be the function from {푎, 푏, 푐, 푑} to {1,2,3} defined by 푓(푎) = 3, 푓(푏) = 2, 푓(푐) = 1, and 푓(푑) = 3. Is f an onto function? Example: Is the function 푓(푥) = 푥2 from the set of integers to the set of integers onto? Example: Determine if each function is a bijection. a. 푓(푥) = 2푥 + 1 © 2020, I. Perepelitsa Page 5 of 9 b. 푓(푥) = 푥2 + 1 Definition: Let 푓 be a bijection from 퐴 to 퐵. Then the inverse of 푓, denoted 푓−1, is the function from 퐵 to 퐴 defined as 푓−1(푦) = 푥 iff 푓(푥) = 푦. No inverse exists unless f is a bijection. Why? © 2020, I. Perepelitsa Page 6 of 9 Example: Let f be the function from {a, b, c} to {1,2,3} such that f(a) = 2, f(b) = 3, and f(c) = 1. Is f invertible and if so what is its inverse? Example: Let 푓: ℤ → ℤ be such that 푓(푥) = 2푥 + 1. Is 푓 invertible, and if so, what is its inverse? Example: Let 푓: ℝ → ℝ be such that 푓(푥) = |푥|. Is 푓 invertible, and if so, what is its inverse? © 2020, I. Perepelitsa Page 7 of 9 Definition: Let 푓: 퐵 → 퐶, 푔: 퐴 → 퐵. The composition of 푓 with 푔, denoted 푓 ∘ 푔 is the function from 퐴 to 퐶 defined by (푓 ∘ 푔)(푥) = 푓(푔(푥)). Example: 푓(푥) = 푥2 and 푔(푥) = 2푥 + 1. Find (푓 ∘ 푔)(푥) and 푔 ∘ 푓(푥). Example: Let g be the function from the set {a, b, c} to itself such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function from the set {a, b, c} to the set {1,2,3} such that f(a) = 3, f(b) = 2, and f(c) = 1. What is the composition of f and g, and what is the composition of g and f. © 2020, I. Perepelitsa Page 8 of 9 Some important functions The floor function, denoted ⌊푥⌋ is the largest integer less than or equal to 푥. The ceiling function, denoted ⌈푥⌉is the smallest integer greater than or equal to 푥. Example: © 2020, I. Perepelitsa Page 9 of 9 Factorial function Definition: 푓: ℕ → ℤ+, denoted by 푓(푛) = 푛! is the product of the first 푛 positive integers. 푓(푛) = 1 ∙ 2 ∙ 3 ⋯ (푛 − 1) ∙ 푛 . 푓(0) = 0! = 1 Example: Find f(2), f(3), f(4). © 2020, I. Perepelitsa .