Maps and functions

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 We often study the properties of a set by comparing it with another set. The "comparison"' is done by a map between the sets. Roughly speaking, a map (also called a function) is a rule to assign an element of a set to an element of another set. 



Let \(A\) and \(B\) be sets. Then we write \(f: A \to B\) to mean that \(f\) is a map from \(A\) to \(B\). That is, for each \(a\in A\), we assign an element \(f(a) \in B\). We often use a "diagram" such as the following:



The mapping of each element \(x \in A\) to \(f(x) \in B\) is written as \(x \mapsto f(x).\)

Example. Let us define a map \(f: \mathbb{Z} \to \mathbb{N}\) by

\[f(x) = 1 + x^2, ~~ (x \in \mathbb{Z}).\]

Note that \(f(x)\in \mathbb{N}\) for any \(x \in \mathbb{Z}\), indeed. □

Definition (domain, codomain)

Let \(A\) and \(B\) be sets, and \(f: A \to B\) be a map. Then, \(A\) and \(B\) are called the domain and codomain of \(f\), respectively. We write \(\text{dom} f = A\) and \(\text{cod} f = B\).

Definition (identity map)

For any set \(A\), the map \(\text{Id}_A: A \to A\) defined by

\[\text{Id}_A(x) = x\]

is called the identity map of \(A\).

Example. \(\text{Id}_{\mathbb{N}}(1) = 1, \text{Id}_{\mathbb{N}}(2) = 2, \text{Id}_{\mathbb{N}}(3) = 3, \cdots\). □

We can also define a function with cases.

Example. We can define \(f: \mathbb{R} \to \{0, 1\}\) by

\[f(x) = \left\{ \begin{array}{cc} 1 & \text{ if $x \in \mathbb{Q}$ },\\ 0 & \text{ otherwise} \end{array}\right.\]

□

Example. The absolute value (modulus) \(|x|\) of a real number \(x\) is a function, \(|\cdot|: \mathbb{R} \to \mathbb{R}\), defined as

\[|x| = \left\{ \begin{array}{cc} x & \text{ if $x \geq 0$},\\ -x & \text{if $x < 0$.} \end{array}\right.\]

For example, \(|3| = 3, |-3| = -(-3) = 3\), etc. □

Definition (image)

Let \(f: A \to B\) be a map. The image of \(f\), denoted \(\text{im}f\), is the set defined by \[\text{im}f = \{f(a) | a \in A\}.\] That is, the set of all possible values of \(f(a)\). Clearly, \(\text{im}f \subset \text{cod}f.\)

Remark. The image of a map \(f\) with the domain \(A\) is also denoted as \(f(A)\). □

Example. Let's define \(f: \mathbb{N} \to \mathbb{R}\) by \(f(x) = x\). Then, \(\text{im}f = \mathbb{N}.\) □

Example. Let \(f: \mathbb{R} \to \mathbb{R}\) be the function \(f(x) = x^2\). Then \(\text{im}f = \{x^2 | x \in \mathbb{R}\} = \{x | x \geq 0\}\), namely, the set of all non-negative real numbers. □

Definition (onto, surjective)

A map \(f: A \to B\) is said to be onto or surjective if \(\text{im}f = \text{cod}f\), that is, 

\[\forall b \in B ~ \exists a \in A ~ (b = f(a)).\]

(Informal translation: every \(b\in B\) comes from \(a \in A\) through \(f\).)

Roughly speaking, the image of a surjective map "covers" the entire codomain.

Example. \(f: \mathbb{Z} \to \mathbb{N}\) defined by \(f(x) = 1 + |x|\) is surjective. However, \(g: \mathbb{Z} \to \mathbb{Z}\) defined by \(g(x) = 1 + |x|\) is not surjective. Note that \(f\) and \(g\) apparently have the same definition, the only difference is their codomains.

Definition (one-to-one, injective)

A map \(f: A \to B\) is said to be one-to-one or injective if, for all \(a, a'\in A\), \(a \neq a'\) implies \(f(a) \neq f(a')\). In a logical form,

\[\forall a, a'\in A ~ (a\neq a' \implies f(a) \neq f(a')).\]

Remark. \(a \neq a' \implies f(a) \neq f(a')\) is equivalent to \(f(a) = f(a') \implies a = a'\) (contrapositive). We can use the latter in the definition above.

Roughly speaking, an injective map maps different elements in \(A\) to different elements in \(B\). If a map is not injective, then different elements in \(A\) can be mapped to the same point in \(B\). That is, there are some different elements \(a \neq a'\) such that \(f(a) = f(a')\).

Example (1). \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = 2x^3 + 1\) is injective. To see this, suppose we have \(f(x) = f(y)\) for some \(x, y\in \mathbb{R}\). Then, \(2x^3 + 1 = 2y^3 + 1\) \(\implies x^3 = y^3\) \(\implies x = y\). □

Example. \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^2\) is not injective (why?). However, \(g: \mathbb{R}_{+} \to \mathbb{R}\) defined by \(g(x) = x^2\), where \(\mathbb{R}_{+} = \{x \in \mathbb{R} | x \geq 0\}\), is injective. □

Definition (bijective)

A map that is both injective and surjective is said to be bijective.

Example. The function in the above Example (1) is bijective. □

We also use nouns such as injection, surjection, and bijection to mean injective, surjective, and bijective maps, respectively.

Definition (Graph (1))

Let \(f: A \to B\) be a map. The graph of \(f\) is the subset of \(A\times B\) defined by
\[\mathrm{graph}(f) = \{(a, b)\mid a\in A, b = f(a)\} (\subset A\times B).\]

This may appear rather abstract. Let's see a concrete example. Let's take \(f(x) = x^2 - 1: \mathbb{R} \to \mathbb{R}\). We can draw the graph of \(y = f(x)\) on the \(x\)-\(y\) plane.  Now, the plane is actually the set \(\mathbb{R}\times \mathbb{R}\). The graph \(y = f(x)\) is a curve in this plane, and hence it is a subset of the plane. The above definition is a generalization of this picture. 

Note the following properties of \(\text{graph}(f)\):
  1. For any \(a \in A\), there exists some \(b \in B\) such that \((a, b)\in \text{graph}(f)\).
  2. If \((a, b), (a, b') \in \text{graph}(f)\), then \(b = b'\).
In short, for any \(a\in A\), there is some \(b\in B\) such that \((a,b) \in \mathrm{graph}(f)\) (part 1) and that \(b\) is unique (part 2).

We can reverse the argument and use these two properties to define the graph.

Definition (Graph (2))

Let \(A\) and \(B\) be sets. Let \(S\) be a subset of \(A\times B\). \(S\) is said to be a graph if it satisfies the following properties:
  1. For any \(a \in A\), there exists some \(b \in B\) such that \((a, b)\in S\).
  2. If \((a, b), (a, b') \in S\), then \(b = b'\).
Note that this definition of the graph does not require the notion of a map. In fact, we can use the notion of the graph to define the map.

Definition (Map)

Let \(A\) and \(B\) be sets, and \(S\) be a graph in the sense of (Graph (2)) above. Then, for any \(a\in A\), there exists a unique \(b\in B\) such that \((a, b) \in S\). We denote this unique \(b\) by \(f_S(a)\). That is, \(f_S(a) = b\). This rule of making correspondence between each \(a\in A\) to \(f_S(a) \in B\) is called a map (induced by \(S\)), denoted 
\[f_S : A \to B.\]

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