Ch. 5: Functions and One-to-One and Onto

What is a bijection?

A function that is one-to-one and onto.

What is another name for a bijection?

A one-to-one correspondence.

Injection is another term for ___?

A one-to-one function.

Surjection is another term for ___?

An onto function.

To say that f: A → B is onto is to say that ___?

Every element of B is the image under f of some element A.

Are these two functions the same? Explain your answer. g = {(x,y)∈ℤ×ℝ| y = 2x + 3} h = {(x,y)∈ℝ×ℝ| y = 2x + 3}.

No, they are not the same. (e, 2e + 3)∈h, but (e, 2e + 3)∉g since e∉ℤ.

f: A → B. Define Ran(f).

Ran(f) ={f(a) | a∈A}.

Function

Suppose F is a relation from A to B. Then F is called a function from A to B if ∀a∈A∃!b∈B((a,b)∈F).

Theorem 5.1.4.

Suppose f and g are functions from A to B. If ∀a∈A(f(a)=g(a)), then f=g.

Theorem 5.2.5.

Suppose f: A → B and g: B → C. As we saw in Theorem 5.1.5., it follows that g∘f: A → C. 1. If f and g are both one-to-one, then so is g∘f. 2. If f and g are both onto, then so is g∘f.

Theorem 5.1.5.

Suppose f: A → B and g: B → C. Then g∘f: A → C, and for any a∈A, the value of g∘f at a is given by the formula (g∘f)(a) = g(f(a)).

Theorem 5.2.3.

Suppose f: A → B. 1. f is one-to-one iff ∀a1∈A∀a2∈A(f(a1)=f(a2) → a1 = a2). 2. f is onto iff Ran(f) = B.

One-to-one

Suppose f: A → B. We will say that f is one-to-one if ¬∃a1∈A∃a2∈A(f(a1)=f(a2) and a1 ≠ a2).

Onto

Suppose f: A → B. We will say that f is onto if ∀b∈B∃a∈A(f(a) = b).

True or False. If f: A → B, then Dom(f) = A and Ran(f) ⊆ B.

True. For f to be a function from A to B, the domain of f must be all of A, but the range of f need not be all of B.

What can we say about f and/or g if we know g∘f is one-to-one?

f is one-to-one.

What do the following terms mean? "The value of f at a," or "the image of a under f," or the result of applying f to a."

f(a).

Suppose f: A → B and C⊆A. Define f↾C.

f↾C = f ∩ (C×B).

Suppose f: A → B and C⊆A, then f↾C is a ____ ?

f↾C: C → B and ∀c∈C(f(c) = (f↾C)(c)). Hint: use existence and uniqueness to prove.

What can we say about f and/or g if we know g∘f is onto?

g is onto.

What can we say about g∘f if f is onto and g is not one-to-one?

g∘f is not one-to-one.

What can we say about g∘f if f is not onto and g is one-to-one?

g∘f is not onto.

The identity relation on A, iA

iA = {(a,a) | a∈A}

Theorem 3.6.1.

∃!P(x) is equivalent to: ∃x(P(x) ∧ ∀y(P(y) → x=y) ∃x∀y(P(y) ↔ y=x) ∃xP(x) ∧ ∀y∀z((P(y) ∧ P(z)) ↔ y=z).