Relations and Functions
Relations and Functions
A relation is any association between elements of one set, called the domain or (less formally) the set
of inputs, and another set, called the range or set of outputs. Some people mistakenly refer to the range
as the codomain, but as we will see, that really means the set of all possible outputs--even values that
the relation does not actually use.For example, if the domain is a set Fruits = {apples, oranges,
bananas} and the codomain is a set Flavors = {sweetness, tartness, bitterness}, the flavors of these
fruits form a relation: we might say that apples are related to (or associated with) both sweetness and
tartness, while oranges are related to tartness only and bananas to sweetness only. (We might disagree
somewhat, but that is irrelevant to the topic of this book.) Notice that "bitterness", although it is one of
the possible Flavors (codomain), is not really used for any of these relationships; so it is not part of the
range {sweetness, tartness}.
Another way of looking at this is to say that a relation is a subset of ordered pairs drawn from the set
of all possible ordered pairs (of elements of two other sets, which we normally refer to as the
Cartesian product of those sets). Formally, R is a relation if R
{(x, y) | x
X, y
Y} for the
domain X and codomain Y.Using the example above, we can write the relation in set notation:
{(apples, sweetness), (apples, tartness), (oranges, tartness), (bananas, sweetness)}.One important kind
of relation is the function.
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A function is a relation that has exactly one output for every possible input in the domain.(Unlike the
codomain, the domain does not necessarily have to include all possible objects of a given type. In fact,
we sometimes intentionally use a restricted domain in order to satisfy some desirable property.) For
example, the relation that we discussed above (flavors of fruits) is not a function, because it has two
possible outputs for the input "apples": sweetness and tartness.The main reason for not allowing
multiple outputs with the same input is that it lets us apply the same function to different forms of the
same thing without changing their equivalence. That is, if x = y, and f is a function with x (or y) in its
domain, then f(x) = f(y). For example, z - 3 = 5 implies that z = 8 because f(x) = x + 3 is a function
defined for all numbers x.The converse, that f(x) = f(y) implies x = y, is not always true. When it is, f is
called a one-to-one or invertible function.
In the above section dealing with functions and their properties, we noted the important property that all
functions must have, namely that if a function does map a value from its domain to its co-domain, it
must map this value to only one value in the co-domain.Writing in set notation, if a is some fixed value:
|{f(x)|x=a}|
{0, 1} The literal reading of this statement is: the cardinality (number of elements) of the
set of all values f(x), such that x=a for some fixed value a, is an element of the set {0, 1}. In other
words, the number of outputs that a function f may have at any fixed input a is either zero (in which
case it is undefined at that input) or one (in which case the output is unique).However, when we
consider the relation, we relax this constriction, and so a relation may map one value to more than one
other value. In general, a relation is any subset of the Cartesian product of its domain and co-domain.
A function is a relationship between two sets of numbers. We may think of this as a mapping; a function
maps a number in one set to a number in another set. Notice that a function maps values to one and
only one value. Two values in one set could map to one value, but one value must never map to two
values: that would be a relation, not a function.For example, if we write (define) a function as: then we
say:If D is a set, we can say which forms a new set, called the range of f. D is called the domain of f,
and represents all values that f takes.In general, the range of f is usually a subset of a larger set. This set
is known as the codomain of a function. For example, with the function f(x)=cos x, the range of f is [-
1,1], but the codomain is the set of real numbers.'f of x equals x squared' and we have and so on. This
function f maps numbers to their squares.Notice that we can have a function that maps a point (x,y) to a
real number, or some other function of two variable.
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