Archetype (mathematics)

The archetype of the element or the one-element subset is the three-element set .

{\ displaystyle 0}

{\ displaystyle \ {0 \} \ subseteq B}

{\ displaystyle \ {2,3,5 \} \ subseteq A}

In mathematics , the archetype is a term associated with images and functions . The archetype of a set under a function is the set of elements that are mapped onto an element in . An element from the definition set of lies in the archetype of if and only if in lies. Thus the archetype of a subset of the target set of a function is a subset of its definition set . Since functions are left total, the archetype corresponds to the definition set, if one considers the entire image set . ${\ displaystyle M}$ ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle M}$ ${\ displaystyle x}$ ${\ displaystyle f}$ ${\ displaystyle M}$ ${\ displaystyle f (x)}$ ${\ displaystyle M}$ ${\ displaystyle B}$ ${\ displaystyle f \ colon A \ to B}$ ${\ displaystyle A}$

definition

Let be a function and a subset of . Then one designates the amount ${\ displaystyle f \ colon A \ to B}$ ${\ displaystyle M}$ ${\ displaystyle B}$

{\ displaystyle f ^ {- 1} (M): = \ left \ {x \ in A \ mid f (x) \ in M ​​\ right \}}

as the archetype of M under f.

An archetype is thus a value of the so-called archetype function, which assigns the archetype to each element of the power set of the target set as an element of the power set of the definition set . ${\ displaystyle M}$ ${\ displaystyle {\ mathcal {P}} (B)}$ ${\ displaystyle B}$ ${\ displaystyle f ^ {- 1} (M)}$ ${\ displaystyle {\ mathcal {P}} (A)}$ ${\ displaystyle A}$

The archetype of a one-element set is also written as ${\ displaystyle M = \ {b \}}$

{\ displaystyle f ^ {- 1} (b): = f ^ {- 1} (\ {b \}) = \ {x \ in A \ mid f (x) = b \}}

and calls it the archetype of b under f. However, this set does not have to be a single element (it can also be empty or contain more than one element).

The archetype of an element is sometimes also called the fiber of the image above this element, especially in connection with fiber bundles .

Examples

For the function ( whole numbers ) with : ${\ displaystyle f \ colon \ mathbb {Z} \ to \ mathbb {Z}}$ ${\ displaystyle f (x) = x ^ {2}}$

{\ displaystyle f ^ {- 1} (4) = \ {2, -2 \}}

{\ displaystyle f ^ {- 1} (0) = \ {0 \}}

{\ displaystyle f ^ {- 1} (3) = \ emptyset}

{\ displaystyle f ^ {- 1} (- 1) = \ emptyset}

{\ displaystyle f ^ {- 1} (\ {1,4 \}) = \ {- 2, -1,1,2 \}}

properties

Injectivity, surjectivity, bijectivity

Under a bijective function , the archetype of each element is (precisely) one element. The mapping that assigns to each element of the (only, thus uniquely determined) element of its original image is called the inverse function of . They are called (also - like the archetype function) with . This can easily lead to misunderstandings if one does not write in more detail for the inverse function (which then clearly distinguishes it from the archetype function ). ${\ displaystyle f \ colon A \ to B}$ ${\ displaystyle B}$ ${\ displaystyle f}$ ${\ displaystyle f ^ {- 1}}$ ${\ displaystyle f ^ {- 1} \ colon B \ to A}$ ${\ displaystyle f ^ {- 1} \ colon {\ mathcal {P}} (B) \ to {\ mathcal {P}} (A)}$

Under an injective function, the archetype of each element is at most one-element (i.e. one-element or empty).

Under a surjective function, the archetype of each element is at least one-element (i.e. not empty).

Set operations and properties

Let it be a function and and be subsets of . Then: ${\ displaystyle f \ colon A \ to B}$ ${\ displaystyle M}$ ${\ displaystyle N}$ ${\ displaystyle B}$

${\ displaystyle f ^ {- 1} (\ emptyset) = \ emptyset}$
${\ displaystyle f ^ {- 1} (B) = A}$
${\ displaystyle f ^ {- 1} (M \ cup N) = f ^ {- 1} (M) \ cup f ^ {- 1} (N)}$
${\ displaystyle f ^ {- 1} (M \ cap N) = f ^ {- 1} (M) \ cap f ^ {- 1} (N)}$
The last two statements (about union and intersection ) can be generalized from two subsets to any families of subsets.

${\ displaystyle f ^ {- 1} (M ^ {\ rm {c}}) = (f ^ {- 1} (M)) ^ {\ rm {c}}}$
The complement of denotes in the respective basic amount . ${\ displaystyle X ^ {\ rm {c}}}$ ${\ displaystyle G \ setminus X: = \ left \ {g \ in G \ mid g \ not \ in X \ right \}}$ ${\ displaystyle X}$ ${\ displaystyle G}$

{\ displaystyle f ^ {- 1} (M \ setminus N) = f ^ {- 1} (M) \ setminus f ^ {- 1} (N)}

{\ displaystyle M \ subseteq N \ Rightarrow f ^ {- 1} (M) \ subseteq f ^ {- 1} (N)}

Image and archetype

Let it be a function, a subset of and a subset of . Then: ${\ displaystyle f \ colon A \ to B}$ ${\ displaystyle M}$ ${\ displaystyle A}$ ${\ displaystyle N}$ ${\ displaystyle B}$

{\ displaystyle f (M) \ subseteq N \ iff M \ subseteq f ^ {- 1} (N),}

that is, there is a Galois connection .

${\ displaystyle M \ subseteq f ^ {- 1} (f (M))}$
If injective , then equality applies. ${\ displaystyle f}$

${\ displaystyle f (f ^ {- 1} (N)) \ subseteq N}$
If surjective , then equality holds. It is sufficient that a subset of the image is from . ${\ displaystyle f}$ ${\ displaystyle N \ subseteq f (A)}$ ${\ displaystyle N}$ ${\ displaystyle \ operatorname {im} f: = f (A) = \ {f (a) \ mid a \ in A \}}$ ${\ displaystyle f}$

Archetype and composition

For any set and any function denote the composition of with . ${\ displaystyle A, B, C}$ ${\ displaystyle f \ colon A \ to B, g \ colon B \ to C}$ ${\ displaystyle g \ circ f \ colon A \ to C}$ ${\ displaystyle g}$ ${\ displaystyle f}$