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Maths › 03. Logic and Foundations ›

10.5 Equivalence

The previous section introduced several formal models of computation and stated that they compute the same class of functions. In this section, we give a more systematic account of this equivalence by presenting explicit constructions that translate computations from one model into another. The central result is that all standard models define exactly the same class of partial functions: $$ f : \mathbb{N}^n \rightharpoonup \mathbb{N}. $$ This class is called...
logic computability equivalence turing machines recursive functions May 2, 2026
Maths › 03. Logic and Foundations ›

1.3 Equivalence

Logical equivalence compares formulas by their truth values under all valuations, rather than by their written form, and it gives a precise way to say that two formulas express the same logical content. Definition 1.17 (Logical Equivalence) Let $A$ and $B$ be formulas. We say that $A$ and $B$ are logically equivalent if: $$ v(A)=v(B) $$ for every valuation $v$. In this case we write: $$ A \equiv B $$...
logic propositional logic equivalence boolean algebra May 2, 2026
Maths › 00. General Mathematics ›

04.3 Isomorphism

4.3 Isomorphism as Sameness Isomorphism is the mathematical notion of structural sameness. Two objects are isomorphic when there is a reversible map between them that preserves the relevant structure. Equality is usually too strict. Two objects can have different representations and still behave identically for the purposes of a theory. For example, the additive group of integers and the additive group of even integers are not equal as sets: $$...
mathematics structure isomorphism equivalence classification May 2, 2026
Maths › 00. General Mathematics ›

01.3 Equality, Identity, and Equivalence

1.3 Equality, Identity, and Equivalence Mathematics depends on precise notions of sameness. The symbol = appears simple, but it carries different meanings depending on context. This section separates three levels: equality, identity, and equivalence. Equality is the strictest notion. Two objects are equal if they are the same object in the underlying system. a = b In set theory, equality is extensional. Two sets are equal when they have exactly...
mathematics foundations equality equivalence structure May 2, 2026