Lecture Discrete structures - Chapter 20: Functions (continue). In this chapter, the following content will be discussed: One-to-one function, onto function, bijective function (one-to-one correspondence), inverse functions. | (CSC 102) Lecture 20 Discrete Structures Previous Lecture Summery Relations and Functions Definition of Function Examples of Functions Functions Today’s Lecture One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Inverse Functions Let F: R → R and G: R → R be functions. Define new functions F + G: R → R: For all x ∈ R, (F + G)(x) = F(x) + G(x) F and G must have same Domains and Codomains. Sum/difference of Functions Theorem: If F: X → Y and G: X → Y are functions, then F = G if, and only if, F(x) = G(x) for all x ∈ X. Equality of Functions Example One-to-One Functions One-to-One Functions 5 6 7 8 9 1 2 3 4 5 The Rule is ‘ADD 4’ Dom (R) = {1, 2, 3, 4, 5} Codomain(R)={5, 6, 7, 8, 9,10} One-to-One Functions Identifying One-to-One functions defined on sets One-to-One Functions on Infinite Sets One-to-One Functions on Infinite Sets One-to-One Functions on Infinite Sets One-to-One Functions on Infinite Sets Onto Functions on Sets Onto Functions on Sets Onto Functions on Sets Identifying Onto Functions Onto Functions on Infinite Sets Onto Functions on Infinite Sets To prove that f is onto, you must prove ∀y ∈ Y, ∃x ∈ X such that f (x) = y. Onto Functions on Infinite Sets There exists real number x such that y = f(x)? Does f really send x to y? Onto Functions on Infinite Sets Onto Functions on Infinite Sets Onto Functions on Infinite Sets One-to-One Correspondence (Bijection) One-to-One Correspondence (Bijection) Inverse Functions Theorem The function F-1 is called inverse function. Finding an Inverse Function The function f : R → R defined by the formula f (x) = 4x − 1, for all real numbers x Theorem Lecture Summery One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Inverse . | (CSC 102) Lecture 20 Discrete Structures Previous Lecture Summery Relations and Functions Definition of Function Examples of Functions Functions Today’s Lecture One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Inverse Functions Let F: R → R and G: R → R be functions. Define new functions F + G: R → R: For all x ∈ R, (F + G)(x) = F(x) + G(x) F and G must have same Domains and Codomains. Sum/difference of Functions Theorem: If F: X → Y and G: X → Y are functions, then F = G if, and only if, F(x) = G(x) for all x ∈ X. Equality of Functions Example One-to-One Functions One-to-One Functions 5 6 7 8 9 1 2 3 4 5 The Rule is ‘ADD 4’ Dom (R) = {1, 2, 3, 4, 5} Codomain(R)={5, 6, 7, 8, 9,10} One-to-One Functions Identifying One-to-One functions defined on sets One-to-One Functions on Infinite Sets One-to-One Functions on Infinite Sets One-to-One Functions on Infinite Sets One-to-One Functions on Infinite Sets Onto Functions on Sets Onto Functions on Sets Onto Functions on Sets Identifying Onto Functions Onto Functions on Infinite Sets Onto Functions on Infinite Sets To prove that f is onto, you must prove ∀y ∈ Y, ∃x ∈ X such that f (x) = y. Onto Functions on Infinite Sets There exists real number x such that y = f(x)? Does f really send x to y? Onto Functions on Infinite Sets Onto Functions on Infinite Sets Onto Functions on Infinite Sets One-to-One Correspondence (Bijection) One-to-One Correspondence (Bijection) Inverse Functions Theorem The function F-1 is called inverse function. Finding an Inverse Function The function f : R → R defined by the formula f (x) = 4x − 1, for all real numbers x Theorem Lecture Summery One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Inverse Functions
