The quotient of a four-element carrier by parity

Goal and scope

X/R={{0,2},{1,3}}, and projection values are whole classes.

A complete finite carrier and parity equivalence; no automatically inherited operation.

Givens and symbols

• calc

• G1 — X={0,1,2,3}; xRy exactly when 2 divides x-y.

• E,O — even and odd classes
• \pi — canonical projection

Rules and sources

• calc · prerequisite — Assume integer divisibility laws, extensional set equality and the definition of a function.
• mko_definition · cited — X/R is the set of all distinct equivalence classes, not a set of selected representatives. The classes cover X and are pairwise disjoint. The canonical projection sends each element to its class. Any quotient operation still needs a separate proof of independence from representative choice. · mko-quotient-set

Main derivation

x\in X

2\mid x-x=0

2\mid x-y

2\mid y-x=-(x-y)

2\mid x-y,\quad2\mid y-z

2\mid x-z=(x-y)+(y-z)

[x]=\{y\in X:2\mid y-x\}

[0]=[2]=\{0,2\}

[1],\ [3]

[1]=[3]=\{1,3\}

\{[0],[1],[2],[3]\}

X/R=\{\{0,2\},\{1,3\}\}

E=\{0,2\},\quad O=\{1,3\}

E\cup O=X,\quad E\cap O=\varnothing

\pi:X\to X/R

\pi(0)=E,\ \pi(1)=O,\ \pi(2)=E,\ \pi(3)=O

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S06, S07 — Outputs are E or O, not numbers 0 or 1; no new quotient operation has been declared.
• S07 · premise_audit · G1, S06 — Both classes are nonempty; coverage and disjointness are checked, not merely the number of classes.
• S06 · premise_audit · G1, S04, S05 — Deduplication removes repeated classes, not distinct elements within a class; {0,1} is only a representative set.
• S05 · premise_audit · G1, S03 — 1 and 3 relate to each other but not to 0 or 2, so no class is missing.
• S04 · premise_audit · G1, S03 — Do not retain only representative 0; 2 belongs to the same class, not another quotient element.
• S03 · premise_audit · G1, S01, S02 — x,y,z must all lie in the same carrier X.
• S02 · premise_audit · G1 — Negating again recovers x-y, so no direction is lost.
• S01 · premise_audit · G1 — The formula covers all four elements without choosing representatives.

Error controls and negative controls

Projection type

The codomain of pi is the set of classes, not selected representatives.

Separate well-definedness

To define f([x])=g(x), verify xRy implies g(x)=g(y).

• X/R={0,1}. → reject — 0 and 1 can be representatives, but the quotient elements are sets E and O.
• [0] and [2] are distinct quotient elements. → reject — Both classes are exactly {0,2}.
• Every function on X descends to the quotient. → reject — f(x)=x differs on 0 and 2, so it cannot define a single value f([0]).

Result and limitations

X/R={{0,2},{1,3}}, and projection values are whole classes.

Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 4172954596b8f050df418adbfe5f100efbfc44e67fa2aa975c7d69dddbead328
• original source SHA-256: 0b6728555127f07d75805ac6a167e55fce6180e7883cf8748e68bfbb3b5aab46
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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