Images像ぞう and preimages逆像ぎゃくぞう of a map写像しゃぞう

A map写像しゃぞう sends elements元げん by a rule. In set theory, however, we often study how whole subsets部分集合ぶぶんしゅうごう move, not only individual elements. The tools for this are the image像ぞう and the preimage逆像ぎゃくぞう.

1Image像ぞう

Let $$f:A\to B$$f:A\to B and $$S\subseteq A$$S\subseteq A. The image像ぞう of $$S$$S is defined by

This is the set集合しゅうごう obtained by applying $$f$$f to the elements元げん of $$S$$S.

In particular, $$f\left(A\right)$$f(A) is the range値域ちいき. Distinguish the codomain終域しゅういき $$B$$B from the range $$f\left(A\right)$$f(A).

2Preimage逆像ぎゃくぞう

Let $$T\subseteq B$$T\subseteq B. The preimage逆像ぎゃくぞう of $$T$$T is defined by

Although the notation $$f^{-1}\left(T\right)$$f^{-1}(T) is used, no inverse map逆写像ぎゃくしゃぞう has to exist. A preimage means “all inputs入力にゅうりょく whose outputs出力しゅつりょく lie in $$T$$T,” so it is a different concept from an inverse map.

3Preimages逆像ぎゃくぞう work well with set operations集合演算しゅうごうえんざん

Preimages逆像ぎゃくぞう preserve unions和集合わしゅうごう, intersections共通部分きょうつうぶぶん, and complements補集合ほしゅうごう.

The reason is that each statement is decided only by whether $$f\left(x\right)$$f(x) belongs to the relevant output-side set.

4Be careful with intersections共通部分きょうつうぶぶん of images像ぞう

Images像ぞう behave well with unions和集合わしゅうごう:

Indeed, $$y\in f(S_1\cup S_2)$$y\in f(S_1\cup S_2) means that $$y=f\left(x\right)$$y=f(x) for some $$x\in S_1\cup S_2$$x\in S_1\cup S_2. Such an $$x$$x lies in $$S_1$$S_1 or $$S_2$$S_2, which is equivalent to $$y\in f\left(S_1\right)\cup f\left(S_2\right)$$y\in f(S_1)\cup f(S_2).

However, for intersections共通部分きょうつうぶぶん, generally only

can be guaranteed. Equality does not always hold.

5Counterexample: images像ぞう do not preserve intersections共通部分きょうつうぶぶん

For a concrete example, let $$A=\{1,2\}$$A=\{1,2\}, $$B=\left\{0\right\}$$B=\{0\}, with $$f\left(1\right)=0$$f(1)=0 and $$f\left(2\right)=0$$f(2)=0. Let $$S_1=\left\{1\right\}$$S_1=\{1\} and $$S_2=\left\{2\right\}$$S_2=\{2\}. Then

so $$f(S_1\cap S_2)=\text{[PARSE ERROR: Undefined("Command("varnothing")")]}$$f(S_1\cap S_2)=\varnothing. But

This happens because $$f$$f is not an injection単射たんしゃ: different inputs入力にゅうりょく collapse to the same output出力しゅつりょく.

6Proposition: equality for images像ぞう of intersections共通部分きょうつうぶぶん

For a map写像しゃぞう $$f:A\to B$$f:A\to B and $$S_1,S_2\subseteq A$$S_1,S_2\subseteq A, one always has

Moreover, if $$f$$f is an injection単射たんしゃ, then for all $$S_1,S_2\subseteq A$$S_1,S_2\subseteq A,

Conversely, if this equality holds for all $$S_1,S_2\subseteq A$$S_1,S_2\subseteq A, then $$f$$f is injective.

For the inclusion, let $$y\in f(S_1\cap S_2)$$y\in f(S_1\cap S_2). There is some $$x\in S_1\cap S_2$$x\in S_1\cap S_2 with $$y=f\left(x\right)$$y=f(x), so $$y\in f\left(S_1\right)\cap f\left(S_2\right)$$y\in f(S_1)\cap f(S_2).

Now suppose $$f$$f is injective and $$y\in f\left(S_1\right)\cap f\left(S_2\right)$$y\in f(S_1)\cap f(S_2). There are $$x_1\in S_1$$x_1\in S_1 and $$x_2\in S_2$$x_2\in S_2 such that $$f\left(x_1\right)=y=f\left(x_2\right)$$f(x_1)=y=f(x_2). Injectivity単射性たんしゃせい gives $$x_1=x_2$$x_1=x_2, so this element lies in $$S_1\cap S_2$$S_1\cap S_2 and $$y\in f(S_1\cap S_2)$$y\in f(S_1\cap S_2).

Finally, suppose the equality holds for all subsets. If $$f\left(x_1\right)=f\left(x_2\right)$$f(x_1)=f(x_2), set $$S_1=\left\{x_1\right\}$$S_1=\{x_1\} and $$S_2=\left\{x_2\right\}$$S_2=\{x_2\}. The set $$f\left(S_1\right)\cap f\left(S_2\right)$$f(S_1)\cap f(S_2) is nonempty, so the assumed equality makes $$f(S_1\cap S_2)$$f(S_1\cap S_2) nonempty. Hence $$S_1\cap S_2$$S_1\cap S_2 is nonempty, which gives $$x_1=x_2$$x_1=x_2. Thus $$f$$f is injective.

7What changes and what is preserved

An image像ぞう carries an input-side subset部分集合ぶぶんしゅうごう to the output side. If a map写像しゃぞう is not an injection単射たんしゃ, different elements元げん may collapse to the same element, so the size of a set and the structure of intersections may change.

A preimage逆像ぎゃくぞう pulls an output-side condition back to an input-side condition. Because the truth of the condition is checked directly through $$f\left(x\right)$$f(x), preimages interact well with set operations集合演算しゅうごうえんざん.

Images interact well with unions和集合わしゅうごう, but they do not always preserve intersections共通部分きょうつうぶぶん. Preimages, by contrast, pull back unions, intersections, and complements as logical conditions, so they have stronger preservation properties.

8Proof supplement: why preimages逆像ぎゃくぞう preserve set operations集合演算しゅうごうえんざん

Let $$f:X\to Y$$f:X\to Y, $$B_1,B_2\subseteq Y$$B_1,B_2\subseteq Y, and $$T\subseteq Y$$T\subseteq Y. For the union identity, rewrite

by rewriting the membership condition:

For the intersection共通部分きょうつうぶぶん identity, replace the logical word “or” by “and” to obtain

For complements補集合ほしゅうごう, explicitly track the ambient sets:

Thus preimages preserve unions, intersections, and complements without assuming injectivity単射性たんしゃせい or surjectivity全射性ぜんしゃせい.

9Next lecture講義こうぎ, exercise link, and summary

An image像ぞう pushes a set forward, while a preimage逆像ぎゃくぞう pulls a condition backward. Always distinguish a preimage from an inverse map逆写像ぎゃくしゃぞう; a preimage is defined even when no inverse map exists.