Lemma·L41
The integer equivalence relation
Pairs of naturals represent differences; two pairs represent the same difference exactly when their cross-sums agree.
Define the relation
on
by
Then
is an equivalence relation on
.
In words
Declare the pair (a, b) equivalent to the pair (c, d) exactly when the cross sums a plus d and c plus b agree: each pair stands for a difference, and two pairs are equivalent exactly when they represent the same difference. This relation is reflexive, symmetric, and transitive.
Never needed: F05 · F10 · F13 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1is a subset of by Separation, hence a relation on (D011).
- 2Reflexive: for , by reflexivity of identity, so .
- 3Symmetric: suppose , i.e. . By symmetry of identity, , which is exactly .
- 4Suppose and , i.e. (i) and (ii). By substitutivity, adding (i) and (ii): By commutativity and associativity of addition, both sides rearrange (they are sums of the same four terms resp. regrouped) to Transitive: cancelling gives , which is exactly .
∎
Remarks
The pair
is meant to represent the difference "
minus
", not yet defined as a number; the condition
is the addition-only way of writing
without subtraction. The integers are defined as the classes of this relation.