Re:Functions are Vectors
by thaumasiotes on Saturday July 29, 2023 @08:56PM
(#36926552)
A bijective function f: A ⟶ B satisfies ∀b∈B ∃a∈A ∀x∈A (f(a) = b ∧ (f(x) = b ⟶ x = a)).
(In fact, this definition is incomplete: it must also be the case that the function is defined over the entire set A; each value in A maps to a value in B.† This is a common assumption to make about functions, but since I violate it earlier by talking about sqrt(x): ℝ ⟶ ℝ, I really should make it explicit. The definition I gave is motivated by the synonymous term "invertible function".)
"All values in the target set have property X" and "Each value in the target set has property X" are exactly equivalent English-language statements.
† "f: A ⟶ B satisfies ∀b∈B ∃a∈A ∀x∈A (f(a) = b ∧ f(x) ∈ B ∧ (f(x) = b ⟶ x = a))", but while that technically works, it feels kludgy.
(In fact, this definition is incomplete: it must also be the case that the function is defined over the entire set A; each value in A maps to a value in B.† This is a common assumption to make about functions, but since I violate it earlier by talking about sqrt(x): ℝ ⟶ ℝ, I really should make it explicit. The definition I gave is motivated by the synonymous term "invertible function".)
"All values in the target set have property X" and "Each value in the target set has property X" are exactly equivalent English-language statements.
† "f: A ⟶ B satisfies ∀b∈B ∃a∈A ∀x∈A (f(a) = b ∧ f(x) ∈ B ∧ (f(x) = b ⟶ x = a))", but while that technically works, it feels kludgy.