7.2 One-to-One and Onto Functions; Inverse Functions

7.2 One-to-One and Onto Functions; Inverse Functions 1 / 1 2 f is called onto (surjective) if f (A) = B. 3 f is called bijective (textbook notation: one-to-one correspondence) if f is both one-to-one and onto.

One-to-one, onto, and bijective functions

Definition Let f : A → B be a function. 1 f is called one-to-one (injective) if a 6= a0 implies f (a) 6= f (a0).

7.2 One-to-One and Onto Functions; Inverse Functions 2 / 1 3 f is called bijective (textbook notation: one-to-one correspondence) if f is both one-to-one and onto.

One-to-one, onto, and bijective functions

Definition Let f : A → B be a function. 1 f is called one-to-one (injective) if a 6= a0 implies f (a) 6= f (a0). 2 f is called onto (surjective) if f (A) = B.

7.2 One-to-One and Onto Functions; Inverse Functions 2 / 1 One-to-one, onto, and bijective functions

Definition Let f : A → B be a function. 1 f is called one-to-one (injective) if a 6= a0 implies f (a) 6= f (a0). 2 f is called onto (surjective) if f (A) = B. 3 f is called bijective (textbook notation: one-to-one correspondence) if f is both one-to-one and onto.

7.2 One-to-One and Onto Functions; Inverse Functions 2 / 1 2 Encoding and decoding functions: Recall from last time: A is the set of all strings of 0’s and 1’s; T is the set of all strings of 0’s and 1’s that consist of consecutive triples of identical bits. The encoding function E : A → T , E(s) =the string obtained from s by replacing each bit of s by the same bit written three times, and the decoding function D : T → A, D(t) =the string obtained from t by replacing each consecutive triple of three identical bits of t by a single copy of that bit. Are E and D one-to-one, onto, bijective functions?

Examples (finite sets)

Examples

1 Let Z3 := {0, 1, 2} and define f : Z3 → Z3 via f (x) = 2x + 1 mod 3. Is f one-to-one? Is it onto? Is it bijective?

7.2 One-to-One and Onto Functions; Inverse Functions 3 / 1 Examples (finite sets)

Examples

1 Let Z3 := {0, 1, 2} and define f : Z3 → Z3 via f (x) = 2x + 1 mod 3. Is f one-to-one? Is it onto? Is it bijective? 2 Encoding and decoding functions: Recall from last time: A is the set of all strings of 0’s and 1’s; T is the set of all strings of 0’s and 1’s that consist of consecutive triples of identical bits. The encoding function E : A → T , E(s) =the string obtained from s by replacing each bit of s by the same bit written three times, and the decoding function D : T → A, D(t) =the string obtained from t by replacing each consecutive triple of three identical bits of t by a single copy of that bit. Are E and D one-to-one, onto, bijective functions?

7.2 One-to-One and Onto Functions; Inverse Functions 3 / 1 2 Let f : R → R defined via f (x) = 3x − 1. Is f (x) one-to-one, onto, bijective? 2 3 Let f : R → R defined via f (x) = x . Is f (x) one-to-one, onto, bijective? How do the answers change if we change the domain of the + function from R to R ?

Examples (infinite sets)

Examples 1 Let f : Z → Z defined via f (n) = 2n. Prove that f is one-to-one but not onto.

7.2 One-to-One and Onto Functions; Inverse Functions 4 / 1 2 3 Let f : R → R defined via f (x) = x . Is f (x) one-to-one, onto, bijective? How do the answers change if we change the domain of the + function from R to R ?

Examples (infinite sets)

Examples 1 Let f : Z → Z defined via f (n) = 2n. Prove that f is one-to-one but not onto. 2 Let f : R → R defined via f (x) = 3x − 1. Is f (x) one-to-one, onto, bijective?

7.2 One-to-One and Onto Functions; Inverse Functions 4 / 1 Examples (infinite sets)

Examples 1 Let f : Z → Z defined via f (n) = 2n. Prove that f is one-to-one but not onto. 2 Let f : R → R defined via f (x) = 3x − 1. Is f (x) one-to-one, onto, bijective? 2 3 Let f : R → R defined via f (x) = x . Is f (x) one-to-one, onto, bijective? How do the answers change if we change the domain of the + function from R to R ?

7.2 One-to-One and Onto Functions; Inverse Functions 4 / 1 Example Find the inverse functions of the bijective functions from the previous examples.

Inverse Functions

Fact If f : A → B is a bijective function then there is a unique function called the inverse function of f and denoted by f −1, such that

f −1(y) = x ⇔ f (x) = y.

7.2 One-to-One and Onto Functions; Inverse Functions 5 / 1 Inverse Functions

Fact If f : A → B is a bijective function then there is a unique function called the inverse function of f and denoted by f −1, such that

f −1(y) = x ⇔ f (x) = y.

Example Find the inverse functions of the bijective functions from the previous examples.

7.2 One-to-One and Onto Functions; Inverse Functions 5 / 1 2 Let E and D be the encoding functions. Find D ◦ E(s) and E ◦ D(s). 3 Important fact: If f : X → Y is a bijective function, then −1 −1 f ◦ f = IY and f ◦ f = IX .

Examples 2 1 Let f : R → R and g : R → R be f (x) = 2x + 1 and g(x) = x . Find g ◦ f (x).

Composition (part of section 7.3)

Definition If f : A → B and g : B → C are functions, then we can define a function g ◦ f : A → C called the composition of g and f such that g ◦ f (x) = g(f (x)).

7.2 One-to-One and Onto Functions; Inverse Functions 6 / 1 3 Important fact: If f : X → Y is a bijective function, then −1 −1 f ◦ f = IY and f ◦ f = IX .

Composition (part of section 7.3)

Definition If f : A → B and g : B → C are functions, then we can define a function g ◦ f : A → C called the composition of g and f such that g ◦ f (x) = g(f (x)).

Examples 2 1 Let f : R → R and g : R → R be f (x) = 2x + 1 and g(x) = x . Find g ◦ f (x). 2 Let E and D be the encoding functions. Find D ◦ E(s) and E ◦ D(s).

7.2 One-to-One and Onto Functions; Inverse Functions 6 / 1 Composition (part of section 7.3)

Definition If f : A → B and g : B → C are functions, then we can define a function g ◦ f : A → C called the composition of g and f such that g ◦ f (x) = g(f (x)).

Examples 2 1 Let f : R → R and g : R → R be f (x) = 2x + 1 and g(x) = x . Find g ◦ f (x). 2 Let E and D be the encoding functions. Find D ◦ E(s) and E ◦ D(s). 3 Important fact: If f : X → Y is a bijective function, then −1 −1 f ◦ f = IY and f ◦ f = IX .

7.2 One-to-One and Onto Functions; Inverse Functions 6 / 1