ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passverified 2026-08-06 (claude-opus-5)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A two-to-one function on a four-element set, its kernel, the quotient, and the canonical factorisation computed explicitly

Example

Write $u : = \emptyset$ , $v : = {\emptyset}$ , $w : = {\emptyset, {\emptyset}}$ and $x : = {{\emptyset}}$ ; these four are pairwise distinct. Put $A : = {u, v} \cup {w, x}$ and $B : = {u, v}$ , and let

$f : = {(u, u), (v, u)} \cup {(w, v), (x, v)},$

so $f : A \to B$ takes the value $u$ twice and the value $v$ twice. Then

$\ker f = {(u, u), (u, v)} \cup {(v, u), (v, v)} \cup {(w, w), (w, x)} \cup {(x, w), (x, x)},$

$A / \ker f = {{u, v}, {w, x}}, f [A] = B,$

and the canonical factorisation $f = ι \circ \overset{ˉ}{f} \circ π$ has $π (u) = π (v) = {u, v}$ , $π (w) = π (x) = {w, x}$ , $\overset{ˉ}{f} ({u, v}) = u$ , $\overset{ˉ}{f} ({w, x}) = v$ , and $ι = Δ_{B}$ .

Facts & Assumptions

Given: $u, v, w, x$ , $A$ , $B$ and $f$ as displayed.

[L1]

$\ker f$ is a set, is a relation on $A$ , and is an equivalence relation on $A$ (For a function $f : A \to B$ the relation $\ker f : = {(a, a^{'}) \in A \times A : f (a) = f (a^{'})}$ is an equivalence relation on $A$ ).

[L2]

there is a function $\overset{ˉ}{f} : A / \sim \to f [A]$ with $\overset{ˉ}{f} ([a]) = f (a)$ for every $a \in A$ , and it is a bijection (Every function $f : A \to B$ factors as $f = ι \circ \overset{ˉ}{f} \circ π$ with $π : A \to A / \ker f$ the surjective quotient map, $\overset{ˉ}{f} : A / \ker f \to f [A]$ the bijection $[a] \mapsto f (a)$ , and $ι : f [A] \to B$ the inclusion injection; $\overset{ˉ}{f}$ is the only function making the factorisation hold).

[L3]

A function is a relation $f$ such that $(a, b) \in f$ and $(a, c) \in f$ imply $b = c$ (A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain).

[L4]

$[a] : = {b \in A : a \sim b} \subseteq A$ (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

[L5]

$A / \sim : = {[a] : a \in A}$ (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

[L6]

$b \in R [A]$ holds if and only if $(a, b) \in R$ for some $a \in A$ (The image $R [A]$ and the preimage $R^{- 1} [B]$ of a set under a relation).

[L7]

${x, y}$ is the set whose elements are exactly $x$ and $y$ , and ${x} : = {x, x}$ (The unordered pair ${x, y}$ and the singleton ${x} = {x, x}$ ).

[L8]

There is exactly one set with no elements, written $\emptyset$ (There is exactly one set with no elements, written $\emptyset$ ).

[L9]

$f$ is surjective (onto) if for every $b \in B$ there is some $x \in A$ with $f (x) = b$ (Injection, surjection, bijection).

[L10]

$z \in a \cup b$ holds if and only if $z \in a$ or $z \in b$ ( $⋃ \emptyset = \emptyset$ , $⋃ {a} = a$ , $⋃ {a, b} = a \cup b$ , $⋂ {a} = a$ , and $⋂ {a, b} = a \cap b$ ).

[L11]

$a \cup b : = ⋃ {a, b}$ (The union $⋃ x$ of a set, and the binary union $a \cup b : = ⋃ {a, b}$ ).

[L12]

$(a, b) = (c, d)$ if and only if $a = c$ and $b = d$ ( $(a, b) = (c, d)$ if and only if $a = c$ and $b = d$ ).

[L13]

$dom R : = {a : \exists b (a, b) \in R}, ran R : = {b : \exists a (a, b) \in R}$ (Relation, $dom R$ , $ran R$ , $fld R$ , and the specialisations "relation from $A$ to $B$ " and "relation on $A$ ").

Verification

technique · direct

1.1

The four sets are pairwise distinct: $u$ has no element; $v$ and $x$ each have exactly one, namely $u$ and $v$ respectively, which differ; and $w$ has both $u$ and $v$ , so it differs from all three. Hence $A$ has exactly four elements and $B$ exactly two.

L7L8L10L11

2.1

$f$ is a function $A \to B$ : its four pairs have distinct first coordinates, so it is single valued; its domain is $A$ and its range is $B$ . It is surjective, since $u = f (u)$ and $v = f (w)$ , so $f [A] = B$ .

L3L6L9L12L13step 1.1

3.1

$\ker f$ is as displayed: a pair $(a, a^{'})$ of elements of $A$ lies in it exactly when $f (a) = f (a^{'})$ , and $f$ takes the value $u$ exactly at $u$ and $v$ , and the value $v$ exactly at $w$ and $x$ ; so the pairs are those with both coordinates in ${u, v}$ or both in ${w, x}$ . By the general lemma it is an equivalence relation on $A$ .

L1L12step 1.1step 2.1

4.1

The classes are $[u] = [v] = {u, v}$ and $[w] = [x] = {w, x}$ , so $A / \ker f$ has exactly the two elements ${u, v}$ and ${w, x}$ , and the quotient map sends $u$ and $v$ to the first and $w$ and $x$ to the second.

L4L5L7step 3.1

5.1

The canonical factorisation is as displayed: $\overset{ˉ}{f}$ sends ${u, v} = [u]$ to $f (u) = u$ and ${w, x} = [w]$ to $f (w) = v$ , and it is a bijection onto $f [A] = B$ ; the inclusion $ι$ is the identity on $B$ because $f [A] = B$ ; and composing gives back $f$ .

L2L6step 2.1step 3.1step 4.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Equivalence relation (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), §3.2 (standard reference, not scraped)
Kernel (set theory) (Wikipedia) (standard reference, not scraped)