Relations and Functions

The One Thing

A Cartesian product forms ordered pairs, a relation selects a subset of those pairs, and a function is a relation that assigns exactly one output in the codomain to every input in the domain. Relations and functions may be represented by ordered pairs, tables, arrow diagrams, equations, or graphs.

Definitions and Results

• Ordered Pair: A pair written as $(a,b)$, where $a$ is the first component and $b$ is the second component. In general, $(a,b)\ne (b,a)$, so order matters.

• Cartesian Product: For sets $A$ and $B$,

$$A\times B=\{(a,b):a\in A,b\in B\}.$$ If $n(A)=m$ and $n(B)=n$, where $A$ and $B$ are finite, then $$n(A\times B)=mn.$$ If either $A$ or $B$ is empty, then $A\times B$ is empty. The Cartesian square of $A$ is $A\times A$. Generally, $A\times B\ne B\times A$, because $(a,b)$ and $(b,a)$ are usually different ordered pairs.

• Relation: A relation from $A$ to $B$ is any subset of $A\times B$:

$$R\subseteq A\times B.$$ The total number of relations from a finite set $A$, with $m$ elements, to a finite set $B$, with $n$ elements, is $$2^{mn}.$$

• Domain: The set of all first components of the ordered pairs in a relation.

• Codomain: The set into which a relation or function maps the elements of its domain. For a function $f:A\to B$, the codomain is $B$.

• Range: The set of all second components, or actual images, occurring in a relation or function. For $f:A\to B$, the range is always a subset of the codomain.

• Image: If $(a,b)$ belongs to a relation or function, then $b$ is the image of $a$.

• Pre-image: If $(a,b)$ belongs to a relation or function, then $a$ is the pre-image of $b$.

• Function: A relation $f$ from $A$ to $B$ is a function, written

$$f:A\to B,$$ if every element of $A$ is associated with exactly one element of $B$. A relation is not a function if any domain element is associated with no element or with more than one element of the codomain.

• Into Function: A function whose range is a proper subset of its codomain.

• Onto Function: A function whose range is equal to its codomain. Equivalently, every element of the codomain is the image of at least one element of the domain.

• One-one Function: A function in which distinct domain elements have distinct images. Equivalently,

$$f(a)=f(b)⟹a=b.$$

• Many-one Function: A function in which two or more distinct domain elements may have the same image.

• Bijective Function: A function that is both one-one and onto.

• Identity Function: A function from a set $A$ to itself defined by

$$f(x)=x$$ for every $x\in A$.

• Constant Function: A function in which every element of the domain has the same image.

• Real Function: A function whose domain and codomain are subsets of the real numbers. The domain consists of allowed input values, and the range consists of the resulting output values.

• Graph of a Function: The set of points

$$(x,f(x))$$ in the Cartesian plane representing the function. The vertical line test states that a graph represents a function of $x$ if every vertical line intersects it at most once.

• Number of Functions: If $A$ and $B$ are finite sets with $n(A)=m$ and $n(B)=n$, then the number of functions from $A$ to $B$ is

$$n^m.$$

• Common Real Functions: These include constant functions, polynomial functions, rational functions, modulus functions, signum functions, and greatest integer functions.

• Modulus Function:

$$|x|=\{\begin{array}{ll}x,& x\ge 0,\\ -x,& x<0.\end{array}$$

• Signum Function:

$$\mathrm{sgn}(x)=\{\begin{array}{ll}1,& x>0,\\ 0,& x=0,\\ -1,& x<0.\end{array}$$

• Greatest Integer Function: Written as $[x]$ or $\lfloor x\rfloor$, it gives the greatest integer less than or equal to $x$.

Worked Methods

Constructing a Cartesian Product

• List each element of $A$.
• Pair each element of $A$ with every element of $B$.
• Write the pairs in the ordered form $(a,b)$, keeping the element from $A$ first.
• If $n(A)=m$ and $n(B)=n$, verify that the product contains $mn$ ordered pairs.
• If either set is empty, conclude that $A\times B=\varnothing$.

The order must be retained: $A\times B$ consists of pairs $(a,b)$, whereas $B\times A$ consists of pairs $(b,a)$. These products are generally different.

Determining a Relation

• Form $A\times B$.
• Select any collection of ordered pairs from $A\times B$.
• Write the selected collection as $R$.
• State that $R\subseteq A\times B$.
• Identify the domain from the first components and the range from the second components.

For finite sets with $m$ and $n$ elements, each of the $mn$ pairs in $A\times B$ may either be selected or not selected. Therefore, the number of possible relations is $$2^{mn}.$$

Testing Whether a Relation Is a Function

• Identify the domain, namely the set of first components.
• Check each domain element.
• Confirm that each one is associated with exactly one element of the codomain.
• If a domain element has no image or more than one image, the relation is not a function.
• If every domain element has exactly one image, the relation is a function $f:A\to B$.

Different domain elements may have the same image; this produces a many-one function and does not violate the definition of a function.

Classifying a Function

• Determine the actual range from the outputs.
• Compare the range with the codomain:

- If the range is a proper subset, the function is into. - If the range equals the codomain, the function is onto.

• Compare images of distinct domain elements:

- If distinct elements always have distinct images, the function is one-one. - If two or more distinct elements have the same image, the function is many-one.

• If the function is both one-one and onto, classify it as bijective.

Counting Relations and Functions

For relations:

• Calculate the number of elements in $A\times B$, namely $mn$.
• Use the fact that each ordered pair may be included or excluded.
• The number of relations is

$$2^{mn}.$$

For functions:

• For each of the $m$ elements of $A$, choose one of the $n$ elements of $B$.
• The number of possible functions is

$$n^m.$$

Finding the Domain and Range of a Real Function

• Identify the expression defining the function.
• Determine the input values for which the expression is defined; these form the domain.
• Determine the resulting output values; these form the range.
• For a graph, use the vertical line test to determine whether it represents a function of $x$.
• For specific real functions, apply the defining rule:

- For $|x|$, use $x$ when $x\ge 0$ and $-x$ when $x<0$. - For $\mathrm{sgn}(x)$, use $1$ for $x>0$, $0$ for $x=0$, and $-1$ for $x<0$. - For $[x]$ or $\lfloor x\rfloor$, select the greatest integer less than or equal to $x$.

Where It Goes Wrong

• Treating $(a,b)$ and $(b,a)$ as the same pair; ordered pairs depend on position.
• Forgetting that a relation must satisfy $R\subseteq A\times B$.
• Calling a relation a function when a domain element has no image or has more than one image.
• Confusing codomain with range: the codomain is specified in $f:A\to B$, whereas the range is determined by the actual outputs.
• Assuming that a function must be one-one; distinct domain elements may have the same image in a many-one function.
• Forgetting the relevant condition when classifying functions: onto requires range equal to codomain, one-one requires distinct images, and bijective requires both.

What Gets Asked

• Construct $A\times B$, $B\times A$, or $A\times A$, and explain why their ordered pairs differ.
• Calculate $n(A\times B)$, including cases where one set is empty.
• List or describe a relation $R\subseteq A\times B$.
• Determine the domain, codomain, range, image, and pre-image of a relation or function.
• Decide whether a given relation is a function.
• Count the number of relations, $2^{mn}$, or functions, $n^m$, between finite sets.
• Classify a function as into, onto, one-one, many-one, or bijective.
• Identify identity and constant functions.
• Determine the domain and range of a real function.
• Apply the vertical line test to a graph.
• Evaluate or describe the modulus function, signum function, and greatest integer function.