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Section 7.1 Definitions and Notation

Definition.

A function from a set \(A\) to a set \(B\) is a rule that associates with each element \(x\) of the set \(A\) exactly one element of the set \(B\text{.}\) A function from \(A\) to \(B\) is also called a mapping from \(A\) to \(B\text{.}\)

Function Notation.

When we work with a function, we usually give it a name. The name is often a single letter, such as \(f\) or \(g\text{.}\) If \(f\) is a function from the set \(A\) to be the set \(B\text{,}\) we will write \(f: A \to B\text{.}\) This is simply shorthand notation for the fact that \(f\) is a function from the set \(A\) to the set \(B\text{.}\) In this case, we also say that \(f\) maps \(A\) to \(B\text{.}\)

Definition.

Let \(f: A \to B\text{.}\) (This is read, “Let \(f\) be a function from \(A\) to \(B\text{.}\)”) The set \(A\) is called the domain of the function \(f\) , and we write \(A = \dom{f}\text{.}\) The set \(B\) is called the codomain of the function \(f\) , and we write \(B = \codom{f}\text{.}\)

If \(a \in A\text{,}\) then the element of \(B\) that is associated with \(a\) is denoted by \(f (a)\) and is called the image of \(a\) under \(f\). If \(f (a) = b\text{,}\) with \(b \in B\text{,}\) then \(a\) is called a preimage of \(b\) under \(f\).

Some Function Terminology.

When we have a function \(f: A \to B\text{,}\) we often write \(y = f(x)\text{.}\) In this case, we consider \(x\) to be an unspecified object that can be chosen from the set \(A\text{,}\) and we would say that \(x\) is the independent variable of the function \(f\) and \(y\) is the dependent variable of the function \(f\text{.}\)

Definition.

Let \(f: A \to B\text{.}\) The set \(\{ f(x) | x \in A \}\) is called the range of the function \(f\) and is denoted by \(\range{f}\text{.}\) The range of \(f\) is sometimes called the image of the function \(f\) (or the image of \(A\) under \(f\) ).

The range of \(f: A \to B\) could equivalently be defined as follows:

\begin{equation*} \range{f} = \{ y \in B | y = f(x) \text{ for some } x \in A \}\text{.} \end{equation*}

Notice that this means that \(\range{f} \subseteq \codom{f}\) but does not necessarily mean that \(\range{f} = \codom{f}\text{.}\) Whether we have this set equality or not depends on the function \(f\text{.}\)

Definition.

Let \(f: A \to B\) be a function from the set \(A\) to the set \(B\text{.}\) The function \(f\) is called an injection provided that

\begin{equation*} \text{for all } x_1, x_2 \in A, \text{ if } x_1 \ne x_2, \text{ then } f (x_1) \ne f (x_2). \end{equation*}

When \(f\) is an injection, we also say that \(f\) is a one-to-one function, or that \(f\) is an injective function.

Notice that the condition that specifies that a function \(f\) is an injection is given in the form of a conditional statement. As we shall see, in proofs, it is usually easier to use the contrapositive of this conditional statement.

Let \(f: A \to B\).

“The function \(f\) is an injection” means that

  • For all \(x_1, x_2 \in A\text{,}\) if \(x_1 \ne x_2\text{,}\) then \(f (x_1) \ne f (x_2)\text{.}\)

  • For all \(x_1, x_2 \in A\text{,}\) if \(f (x_1) = f (x_2)\text{,}\) then \(x_1 = x_2\text{.}\)

“The function \(f\) is not an injection” means that

  • There exist \(x_1, x_2 \in A\) such that \(x_1 \ne x_2\) and \(f(x_1) = f(x_2)\text{.}\)

Definition.

Let \(f: A \to B\) be a function from the set \(A\) to the set \(B\text{.}\) The function \(f\) is called a surjection provided that the range of \(f\) equals the codomain of \(f\text{.}\) This means that

For every \(y \in B\text{,}\) there exists an \(x \in A\) such that \(f(x) = y\text{.}\)
When \(f\) is a surjection, we also say that \(f\) is an onto function or that \(f\) maps \(A\) onto \(B\). We also say that \(f\) is a surjective function.

One of the conditions that specifies that a function \(f\) is a surjection is given in the form of a universally quantified statement, which is the primary statement used in proving a function is (or is not) a surjection.

Let \(f: A \to B\).

“The function \(f\) is an surjection” means that

  • \(\range{f} = \codom{f} = B\text{;}\) or

  • For every \(y \in B\text{,}\) there exists an \(x \in A\) such that \(f(x) = y\text{.}\)

“The function \(f\) is not an surjection” means that

  • \(\range{f} \ne \codom{f}\text{;}\) or

  • There exists a \(y \in B\) such that for all \(x \in A\text{,}\) \(f(x) \ne y\text{.}\)

One last definition.

Definition.

A bijection is a function that is both an injection and a surjection. If the function \(f\) is a bijection, we also say that \(f\) is one-to-one and onto and that \(f\) is a bijective function.