Last Update: 1 December 2010
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Then can define R_{i} ⊆ S × C such that:
Let = ⊆ N×N be defined recursively as follows:
Then: ≤ =def < ∪ =
tval(A)=tval(B) → tval(B)=tval(A)
∴ A≡B →
B≡A
(tval(A)=tval(B) ∧ tval(B)=tval(C)) →
tval(A)=tval(C)
∴ (A≡B ∧ B≡C) → A≡C
QED
Then:
{a′ ∈ A | a′ ∼ a}
Let P = {[a_{1}]_{∼}, …, [a_{n}]_{∼}} be the set of all equivalence classes of elements of A under ∼.
Then ∪_{i}[a_{i}]_{∼} = A.
∴ They are jointly exhaustive.
And (∀i, j)[[a_{i}]_{∼} ∩ [a_{j}]_{∼} = ∅
∴ They are mutually exclusive.
∴ P is a partition of A.
QED.
(∀ partition {A_{1},…,A_{n}} of A)(∃ ∼ that is an equivalence relation on A]