Lemma·L55
The rational equivalence relation
Pairs of integers (with nonzero second coordinate) represent fractions; two pairs represent the same fraction exactly when they cross-multiply equal.
Define the relation
on
by
Then
is an equivalence relation on
.
In words
Pair up two integers, the second nonzero, to represent their ratio; declare two such pairs equivalent when they cross-multiply to equal products: this happens exactly when they represent the same fraction. This relation is reflexive, symmetric, and transitive.
Never needed: F10 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1is a subset of by Separation, hence a relation (D011).
- 2Reflexive: by reflexivity, so .
- 3Symmetric: means ; by symmetry, , which is exactly .
- 4Transitive: suppose and , i.e. (i) and (ii). Multiplying (i) by and (ii) by (substitutivity): By commutativity and associativity of on , , so Rearranging both sides (commutativity/associativity again) to factor out : . Since (as ), cancellation gives , which is exactly .
∎
Remarks
The pair
is meant to represent the fraction "
over
"; the condition
is the division-free way of writing
. The rationals are defined as the classes of this relation. Transitivity genuinely needs cancellation, unlike the naturals-based construction of the integers which only needed additive rearrangement: this is where
having no zero divisors earns its keep.