Discrete Structures Chapter 4: Functions

No (because then you would have a value you could plug in that would give no output whatsoever.)

If any element of the domain of f is unmapped (does not have an arrow coming from it) can f be a function?

well-defined

If f maps an element of the domain to zero elements or more than one element of the target, then f is not ___.

No, because an element in the domain is mapped to two different elements in the target.

(see image) is f a function? If not, explain why.

Yes.

(see image) is g a function? If not, explain why.

function

A ___ f that maps elements of a set X to elements of a set Y, is a subset of X × Y such that for every x ∈ X, there is exactly one y ∈ Y for which (x, y) ∈ f.

one-to-one correspondence.

A bijection is also called a ___

bijection

A bijective function is called a ___.

one-to-one

A function f: X → Y is ___ or injective if x1 ≠ x2 implies that f(x1) ≠ f(x2). That is, f maps different elements in X to different elements in Y.

onto

A function f: X → Y is ___ or surjective if the range of f is equal to the target Y. That is, for every y ∈ Y, there is an x ∈ X such that f(x) = y.

bijective

A function is ___ if it is both one-to-one and onto.

co-domain

An alternate word for target that is sometimes used is ___.

range (In other words, y is in the range if there's an x in the domain that is mapped to y.)

For function f: X → Y, an element y is in the ___ of f if and only if there is an x ∈ X such that (x, y) ∈ f.

important ( so f ο g is not the same as g ο f )

Generally, the order in which the functions are applied is ___

8 ( formula: 2^x where x = the exponent )

How many elements are in the following set? {0, 1}^3

inverse

If a function f: X → Y is a bijection, then the ___ of f is obtained by exchanging the first and second entries in each pair in f. The inverse of f is denoted by f^-1

arrow diagram

In an ___ for a function f, the elements of the domain X are listed on the left and the elements of the target Y are listed on the right. There is an arrow from x ∈ X to y ∈ Y if and only if (x, y) ∈ f.

associative

It is possible to compose more than two functions. Composition is ___, so the order in which one composes the functions does not matter

one arrow

Since f is a function, each x ∈ X has exactly one y ∈ Y such that (x, y) ∈ f, which means that in the arrow diagram for a function, there is exactly ___ pointing out of every element in the domain.

Identity Function

The ___ always maps a set onto itself and maps every element onto itself.

floor ( floor: R → Z, where floor(x) = the largest integer y such that y ≤ x. The floor function comes up so often in mathematics that it has its own notation: floor(x) = |_x_| )

The ___ function maps a real number to the nearest integer in the downward direction.

ceiling

The ___ function rounds a real number to the nearest integer in the upward direction.

inverse

The exponential function is one-to-one and onto, and therefore has an ___.

already known

The fact that the domain and target of a bijection have the same size may seem simple but this fact turns out to be extremely powerful. One way to count the elements in a set is to define a bijection between that set and another set whose size is ___ ___. Counting the elements in a set is a fundamental part of discrete probability, an important tool in many areas of science.

exponential

The logarithm function is the inverse of the ___ function.

base

The parameter b is called the ___ of the logarithm in the expression log₆ y.

composition

The process of applying a function to the result of another function is called ___.

domain target

The set X is called the ___ of f, and the set Y is the ___ of f.

equal

Two functions, f and g, are ___ if f and g have the same domain and target, and f(x) = g(x) for every element x in the domain. The notation f = g is used to denote the fact that functions f and g are equal.

|D| ≥ |T| |D| ≤ |T| |D| = |T|

When the domain and target are finite sets, it is possible to infer information about their relative sizes based on whether the function is one-to-one or onto. If f: D → T is onto, then for every element in the target, there is at least one element in the domain: ??? If f: D → T is one-to-one, then every element in the domain maps to a unique element in the target: ??? If f: D → T is a bijection, then f is one-to-one and onto: |D| ≤ |T| and |D| ≥ |T|, which implies that ???.

Divide-and-conquer

___ is a common strategy in computer science in which a problem is solved for a large set of items by dividing the set of items into two evenly sized groups, solving the problem on each half and then combining the solutions for the two halves.

function

f: X → Y is the notation to express the fact that f is a ___ from X to Y.

exponential function

see image