Basics of maps写像しゃぞう

1Introduction

The first question for a map写像しゃぞう is whether each input入力にゅうりょく determines a unique output出力しゅつりょく. A map is a special kind of relation関係かんけい, but it has stronger conditions than an arbitrary relation.

This condition is what makes the notation $$f\left(a\right)$$f(a) meaningful. If one $$a$$a had several outputs, $$f\left(a\right)$$f(a) would not be a single value値あたい. If it had no output, $$f\left(a\right)$$f(a) would be undefined.

What a map requires is exactly one output for each input. Different inputs may have the same output. The additional condition that different inputs have different outputs will later be called injectivity単射たんしゃ.

2Terms and definition

For sets集合しゅうごう $$A,B$$A,B, a map写像しゃぞう $$f:A\to B$$f:A\to B is a rule assigning to each $$a\in A$$a\in A a unique element $$b\in B$$b\in B.

The set $$A$$A is the domain定義域ていぎいき, and $$B$$B is the codomain終域しゅういき. The assigned value値あたい is written $$f\left(a\right)$$f(a). The range値域ちいき, or image像ぞう, is the set of outputs出力しゅつりょく that actually appear:

Always $$f\left(A\right)\subseteq B$$f(A)\subseteq B.

3Images像ぞう and preimages逆像ぎゃくぞう

For a subset部分集合ぶぶんしゅうごう $$S\subseteq A$$S\subseteq A, the image像ぞう of $$S$$S is

For a subset $$T\subseteq B$$T\subseteq B, the preimage逆像ぎゃくぞう is

The notation $$f^{-1}\left(T\right)$$f^{-1}(T) does not assume that an inverse map逆写像ぎゃくしゃぞう exists. A preimage is defined for every map写像しゃぞう.

4Strategy

To check whether a relation is a map写像しゃぞう, separate two requirements.

1. For every $$a\in A$$a\in A, at least one output出力しゅつりょく exists.
2. For every $$a\in A$$a\in A, at most one output exists.

If both hold, each input has exactly one output, and the relation is a map. From the relation関係かんけい viewpoint, $$f$$f is a subset部分集合ぶぶんしゅうごう of $$A\times B$$A\times B such that for every $$a\in A$$a\in A there is exactly one $$b$$b with $$(a,b)\in f$$(a,b)\in f.

5Intuitive explanation

A map写像しゃぞう can be pictured as drawing one arrow from each element元げん on the input side to the output side. The defining condition is that exactly one arrow leaves every input入力にゅうりょく.

An element on the output side may have no incoming arrows, or it may have several incoming arrows. That is not a problem for being a map. Additional restrictions on these incoming arrows lead to injections単射たんしゃ and surjections全射ぜんしゃ.

In the machine picture, the assignment from inputs to outputs may vary. What remains fixed is that every element of the domain定義域ていぎいき is processed exactly once and that every resulting output lies in the codomain終域しゅういき.

6Precise explanation

For a map写像しゃぞう $$f:A\to B$$f:A\to B, images像ぞう and preimages逆像ぎゃくぞう go in opposite directions. An image sends a subset部分集合ぶぶんしゅうごう of the $$A$$A side to the $$B$$B side:

A preimage sends a subset of the $$B$$B side back to the $$A$$A side:

A preimage is defined even when the map is neither an injection単射たんしゃ nor a surjection全射ぜんしゃ.

7Worked example: distinguish codomain終域しゅういき and range値域ちいき

7.1Problem

Define $$f:\{1,2,3\}\to \{a,b,c,d\}$$f:\{1,2,3\}\to\{a,b,c,d\} by $$f\left(1\right)=a$$f(1)=a, $$f\left(2\right)=b$$f(2)=b, and $$f\left(3\right)=b$$f(3)=b. Find the codomain終域しゅういき and the range値域ちいき.

7.2Explanation

The codomain終域しゅういき is the set集合しゅうごう specified in advance as part of the type of the map写像しゃぞう. Therefore the codomain is $$\{a,b,c,d\}$$\{a,b,c,d\}.

The range値域ちいき is the set of outputs出力しゅつりょく that actually occur as values. In this example only $$a$$a and $$b$$b occur, so

The elements $$c,d$$c,d are in the codomain, but not in the range.

When checking a proposed map, do not rely only on a table or arrow diagram: identify the domain定義域ていぎいき, codomain, and assignment rule explicitly. If some input has no value or has two values, the proposed relation is not a map.

8What changes and what is preserved

Injectivity単射性たんしゃせい means that different inputs入力にゅうりょく are sent to different outputs出力しゅつりょく. Surjectivity全射性ぜんしゃせい means that every element of the codomain終域しゅういき is the value値あたい of at least one input, equivalently $$f\left(A\right)=B$$f(A)=B. The following table compares how changing the domain定義域ていぎいき or codomain affects these properties.

Enlarging or narrowing the codomain終域しゅういき can change whether the same assignment rule is surjective全射ぜんしゃ. The image像ぞう, by contrast, is determined by the domain and the assignment rule.

There is one necessary qualification when narrowing the codomain: the new codomain must still contain the range値域ちいき $$f\left(A\right)$$f(A). If an element of $$f\left(A\right)$$f(A) is removed, some input value lies outside the declared codomain, so the rule no longer defines a map of that type.

9How to check and related links

• If every input入力にゅうりょく has exactly one output出力しゅつりょく, it is a map写像しゃぞう.
• If one input has multiple outputs, it is not a map.
• The codomain終域しゅういき is the declared set of possible outputs.
• The range値域ちいき is the set of outputs actually reached.
• A preimage逆像ぎゃくぞう is defined even when no inverse map逆写像ぎゃくしゃぞう exists.

10Exercise links