19 One-to-one, onto and inverse functions, Composition of functions

Suppose F: X → Y is a one-to-one correspondence; that is, suppose F is one-to-one and onto. Then there is a function F^−1: Y → X that is defined as follows: Given any element y in Y, F^−1(y) = that unique element x in X such that F(x) equals y. In other words, F^−1(y) = x ⇔ y = F(x).

*Inverse function* Suppose F: X → Y is a one-to-one correspondence; that is, suppose F is one-to-one and onto. Then there is a function F^−1: Y → X that is defined as follows: ... In other words, F^−1(y) = __ ⇔ _________

A one-to-one correspondence (or bijection) from a set X to a set Y is a function F: X → Y that is both one-to-one and onto. Thus, a function that is one-to-one and onto sets up a pairing between the elements of X and the elements of Y that matches each element of X with exactly one element of Y and each element of Y with exactly one element of X

A one-to-one correspondence (or _______) from a set X to a set Y is a function F: X → Y that is both ___________________

If f : X → Y and g: Y → Z are both one-to-one functions, then g ◦ f is one-to-one.

If f : X → Y and g: Y → Z are both one-to-one functions, then g ◦ f is ________

If f : X → Y and g: Y → Z are both onto functions, then g ◦ f is onto.

If f : X → Y and g: Y → Z are both onto functions, then g ◦ f is _____

Let f : X → Y ′ and g: Y → Z be functions with the property that the range of f is a subset of the domain of g. Define a new function g ◦ f : X → Z as follows: (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read "g circle f " and g( f (x)) is read "g of f of x." The function g ◦ f is called the *composition* of f and g.

Let f : X → Y ′ and g: Y → Z be functions with the property that the range of f is a subset of the domain of g. Define a new function g ◦ f : X → Z as follows: (g ◦ f )(x) = g( f (x)) for all x ∈ X, where g ◦ f is read "g circle f " and g( f (x)) is read "g of f of x." The function g ◦ f is called the *composition* of f and g.

A function F: X → Y is not one-to-one ⇔ ∃ elements x1 and x2 in X with F(x1) = F(x2) and x1 != x2.

To obtain a precise statement of what it means for a function not to be one-to-one, take the negation of one of the equivalent versions of the definition above:

When a function is onto or surjective, its range is equal to its co-domain.

When a function is onto or _______, if ________________

if no two arrows that start in the domain point to the same element of the co-domain then the function is called one-to-one or injective

if *no* two arrows that start in the domain point to the same element of the co-domain then the function is called _____ or ______

one-to-one = injective

one-to-one = _____

the composition of any function with an identity function equals the function.

the composition of any function with an identity function equals _______

the composition of f and f ^−1 sends each element to itself. So by definition of the identity function, f ^−1 ◦ f = Ix More generally, the composition of any function with its inverse (if it has one) is an identity function

the composition of f and f ^−1 sends each element ______. So by definition of the identity function, f ^−1 ◦ f = ____