Applied Discrete Structures

The reader should note that a function $f$ is a relation from $A$ into $B$ with two important restrictions:

It is customary to use a different system of notation for functions than the one we used for relations. If $f$ is a function from the set $A$ into the set $B\text{,}$ we will write $f:A\to B\text{.}$

The reader is probably more familiar with the notation for describing functions that is used in basic algebra or calculus courses. For example, $y=\frac{1}{x}$ or $f(x)=\frac{1}{x}$ both define the function $\{(x,\frac{1}{x})|x\in \mathbb{R},x\ne 0\}\text{.}$ Here the domain was assumed to be those elements of $\mathbb{R}$ whose substitutions for $x$ make sense, the nonzero real numbers, and the codomain was assumed to be $\mathbb{R}\text{.}$ In most cases, we will make a point of listing the domain and codomain in addition to describing what the function does in order to define a function.

The terms mapping, map, and transformation are also used for functions.

The notation used for sets of functions makes sense in light of Exercise 5.

One way to imagine a function and what it does is to think of it as a machine. The machine could be mechanical, electronic, hydraulic, or abstract. Imagine that the machine only accepts certain objects as raw materials or input. The possible raw materials make up the domain. Given some input, the machine produces a finished product that depends on the input. The possible finished products that we imagine could come out of this process make up the codomain.

In Example 7.1.6, the image of 2 under $f$ is 4; that is, $f(2)=4\text{.}$ In Example 7.1.2, the image of $-1$ under $s$ is 1; that is, $s(-1)=1\text{.}$

Note that the range of a function is a subset of its codomain. $f(X)$ is also read as “the image of the set $X$ under the function $f$” or simply “the image of $f\text{.}$”

In Example 7.1.2, $s(A)=\{0,1,4\}\text{.}$ Notice that 2 and 3 are not images of any element of $A\text{.}$ In addition, note that both 1 and 4 are related to more than one element of the domain: $s(1)=s(-1)=1$ and $s(2)=s(-2)=4\text{.}$ This does not violate the definition of a function. Go back and read the definition if this isn’t clear to you.

In Example 7.1.4, the range of $L$ is equal to its codomain, $\mathbb{R}\text{.}$ If $b$ is any real number, we can demonstrate that it belongs to $L(\mathbb{R})$ by finding a real number $x$ for which $L(x)=b\text{.}$ By the definition of $L\text{,}$ $L(x)=3x\text{,}$ which leads us to the equation $3x=b\text{.}$ This equation always has a solution, $\frac{b}{3}\text{;}$ thus $L(\mathbb{R})=\mathbb{R}\text{.}$

The formula that we used to describe the image of a real number under $L\text{,}$ $L(x)=3x\text{,}$ is preferred over the set notation for $L$ due to its brevity. Any time a function can be described with a rule or formula, we will use this form of description. In Example 7.1.2, the image of each element of $A$ is its square. To describe that fact, we write $s(a)=a^2$ ($a\in A$), or $S:A\to B$ defined by $S(a)=a^2\text{.}$

There are many ways that a function can be described. Many factors, such as the complexity of the function, dictate its representation.

If the domain of a function is the Cartesian product of two sets, then our notation and terminology changes slightly. For example, consider the function $G:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ defined by $G\left((n_1,n_2)\right)=n_1^2+n_2^2-n_1{n}_2+10\text{.}$ For this function, we would drop one set of parentheses and write $G(4,2)=22\text{,}$ not $G((4,2))=22\text{.}$ We call $G$ a function of two variables. From one point of view, this function is no different from any others that we have seen. The elements of the domain happen to be slightly more complicated. On the other hand, we can look at the individual components of the ordered pairs as being separate. If we interpret $G$ as giving us the cost of producing quantities of two products, we can imagine varying $n_1$ while $n_2$ is fixed, or vice versa.

The same observations can be made for function of three or more variables.

There are several ways to define a function in Sage. The simplest way to implement $f$ is as follows.

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f(x)=x^2

f

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[f(4),f(1.2)]

Sage is built upon the programming language Python, which is a strongly typed language and so you can’t evaluate expressions such as f('Hello'). However a function such as $f\text{,}$ as defined above, will accept any type of number, so a bit more work is needed to restrict the inputs of $f$ to the integers.

A second way to define a function in Sage is based on Python syntax.

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def fa(x):

    return x^2

#end of definition - now we test it:

[fa(2),fa(1.2)]

We close this section with two examples of relations that are not functions.