Definition D1
A relation from a set X to Y is a subset of X × Y, that is, a collection of ordered pairs (a, b).
Proof.
A relation carries no restriction. Pairing one a with several b is still a relation.convention
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Definition D2
A relation f is a function when, for each a ∈ X, there exists exactly one b with (a, b) ∈ f.
Proof.
This condition does not follow from D1. Imposing F as a requirement is the definition of a function.D1, F
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Definition D3
f(a) means the b with (a, b) ∈ f.
Proof.
The function-value notation points to one such b.convention
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Definition D4
f⁻¹ is the collection of the reversed pairs of f: {(b, a) : (a, b) ∈ f}.
Proof.
The reversal is again a collection of ordered pairs, hence a relation. Whether it is a function is a separate question.D1
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Definition D5
f is injective when f(a₁) = f(a₂) implies a₁ = a₂.
Proof.
It means that distinct inputs do not share the same output.convention
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Definition D6
The range f(X) of f is {f(a) : a ∈ X}.
Proof.
For each element b of the range there is at least one a with f(a) = b.convention
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Lemma L1
If a relation f contains both (a, b) and (a, c) with b ≠ c, the notation f(a) fails to point to any value.
Proof.
(1)Suppose (a, b) ∈ f and (a, c) ∈ f with b ≠ c.assumption
(2)f(a) = b(1), D3
(3)f(a) = c(1), D3
(4)b = f(a) and f(a) = c, so b = c(2), (3), A2
(5)b = c and b ≠ c cannot hold together.(1), (4), A3
(6)Hence D3 fails to assign a value to f(a).(5), D3
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Theorem T1
The relations for which the notation f(a) has a value are exactly the relations satisfying D2.
Proof.
(1)Suppose f satisfies D2. For each a there is exactly one b with (a, b) ∈ f.D2
(2)Hence f(a) points to that one b.(1), D3
(3)Conversely, suppose f does not satisfy D2. For some a, such a b does not exist or there are two or more.D2
(4)If there is no b, f(a) has nothing to point to.(3), D3
(5)If there are two or more, f(a) has no value.(3), L1
(6)Hence the notation f(a) requires D2.(2), (4), (5)
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Theorem T2
For a > 0 the equation x² = a has two solutions, so to use the symbol √a as a function, one of the two must be chosen.
Proof.
(1)If a > 0, then x² = a has two distinct solutions.P1
(2)If we set '√a is the x with x² = a', one a is paired with two distinct values.(1), D1
(3)Then the notation √a has no value.(2), L1
(4)To use √ as a function, D2 must be satisfied.T1
(5)Hence one of the two must be chosen, and the nonnegative one is chosen.(3), (4), P2
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Theorem T3
f⁻¹ is a function on the range f(X) if and only if f is injective.
Proof.
(1)If b ∈ f(X), there is an a with f(a) = b, and (a, b) ∈ f, so (b, a) ∈ f⁻¹.D6, D3, D4
(2)Suppose f is injective.assumption
(3)If (b, a₁), (b, a₂) ∈ f⁻¹, then (a₁, b), (a₂, b) ∈ f.D4
(4)Then f(a₁) = b and b = f(a₂), so f(a₁) = f(a₂).(3), D3, A2
(5)a₁ = a₂(2), (4), D5
(6)By (1) and (5), each b ∈ f(X) corresponds to exactly one element, so f⁻¹ is a function.(1), (5), D2
(7)Conversely, suppose f⁻¹ is a function on f(X).assumption
(8)If f(a₁) = f(a₂) = b, then (b, a₁), (b, a₂) ∈ f⁻¹.D3, D4
(9)b corresponds to only one element, so a₁ = a₂, and f is injective.(7), (8), D2, D5
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Corollary C
A collection G of points in the coordinate plane is the graph of a function if and only if every vertical line x = a meets G in exactly one point.
Proof.
(1)The vertical line x = a is the set of all points of the form (a, y).convention
(2)(a, y) and (a, y′) are the same point only when y = y′.A1
(3)Hence the number of points where the vertical line meets G equals the number of y with (a, y) ∈ G.(1), (2)
(4)G being the graph of a function means each a has exactly one such y.D1, D2
(5)Hence the two statements are equivalent.(3), (4)
(6)The vertical line test is not a rule discovered in a picture but D2 transcribed onto the coordinate plane.(5), D2
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