Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 3

Sets

Georg Cantor (1845-1918) is the creator of set theory.

David Hilbert (1862-1943) has called set theory a paradise, from where mathematicians should never be expelled.

Mathematical structures like numbers are described as sets. A set is a collection of distinct objects which are called the elements of the set. By distinct we mean that it is clear which objects are considered to be equal and which are considered to be different. The containment of an element ${}x$ to a set ${}M$ is expressed by

${}x\in M\,,$

the noncontainment by

${}x\notin M\,.$

For every element, exactly one of these possibilities holds. For example, we have ${}{\frac {3}{7}}\notin \mathbb {N}$ and ${}{\frac {3}{7}}\in \mathbb {Q}$. An important principle for sets is the principle of extensionality, i.e. a set is determined by the elements it contains, beyond that it bears no further information. In particular, two sets coincide if they contain the same elements.

The set which does not contain any element is called the empty set and is denoted by

$\emptyset .$

A set ${}N$ is called a subset of a set ${}M$ if every element from ${}N$ does also belong to ${}M$. For this relation we write ${}N\subseteq M$ (some people write ${}N\subset M$ for this). One also says that the inclusion ${}N\subseteq M$ holds. For the number sets, the inclusions

${}\mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \,$

hold. The subset relation ${}N\subseteq M$ is a statement using for all, as it makes a claim about all elements from ${}N$. If we want to show ${}N\subseteq M$, then we have to show for an arbitrary element ${}x\in N$ that also the containment ${}x\in M$ holds. In order to show this, we are only allowed to use the property ${}x\in N$. For us, sets will be either number sets or sets constructed from such number sets. A set is called finite if its elements may be counted by the natural numbers ${}1,2,3,\ldots ,n$ for a certain ${}n\in \mathbb {N}$. In this case, the number ${}n$ is called the number (or the cardinality) of the set.

Possible descriptions for sets

There are several ways to describe a set. The easiest one is to just list the elements of the set, here the order of the listing is not important. For finite sets, this is possible; however, for infinite sets, one has to describe a "construction rule“ for the elements.

The most important set given by an infinite listing is the set of natural numbers

${}\mathbb {N} =\{0,1,2,3,\ldots \}\,.$

Here, a certain set of numbers is described by the first elements in the hope that this indicates how the listing goes on and which numbers belong to the set. An important point is that ${}\mathbb {N}$ is not a set of certain digits, but the set of values represented by these digits or sequences of digits. For a natural number, there are many possibilities to represent it, the decimal expansion is just one of them.

We discuss now the description of sets by properties. Let a set ${}M$ and a certain property ${}E$ (a predicate) be given, such that the property can be applied to the elements of ${}M$. Hence, for the property ${}E$, we have in ${}M$ the subset consisting of all the elements from ${}M$ which fulfil this property. We write for this subset given by ${}E$

${}{\left\{x\in M\mid E(x)\right\}}={\left\{x\in M\mid x{\text{ fulfils property }}E\right\}}\,.$

This only works for such properties for which the statement ${}E(x)$ is well-defined for every ${}x\in M$. If one introduces such a subset, then one gives a name to it, which often reflects the name of the property, like

${}E={\left\{x\in \mathbb {N} \mid x{\text{ is even}}\right\}}\,,$

${}O={\left\{x\in \mathbb {N} \mid x{\text{ is odd}}\right\}}\,,$

${}S={\left\{x\in \mathbb {N} \mid x{\text{ is a square number}}\right\}}\,,$

${}{\mathbb {P} }={\left\{x\in \mathbb {N} \mid x{\text{ is a prime number}}\right\}}\,.$

For the sets occurring in mathematics, a multitude of mathematical properties is relevant, and, therefore, there is a multitude of relevant subsets. But also in the sets of everyday life like the set ${}C$ of the students in a course, there are many important properties which determine certain subsets, like

${}O={\left\{x\in C\mid x{\text{ lives in Osnabr}}{\ddot {\rm {u}}}{\text{ck}}\right\}}\,,$

${}P={\left\{x\in C\mid x{\text{ studies physics}}\right\}}\,,$

${}D={\left\{x\in C\mid x{\text{ has birthday in December}}\right\}}\,.$

The set ${}C$ itself is also given by a property, since

${}C={\left\{x\mid x{\text{ is a student in the course}}\right\}}\,.$

Set operations

Similar to the construction of new statements from given statements by connecting them with logical connectives, there are operations to make new sets from old ones. The most important operations are the following:

1. Union

   ${}A\cup B:={\left\{x\mid x\in A{\text{ or }}x\in B\right\}}\,.$

2. Intersection

   ${}A\cap B:={\left\{x\mid x\in A{\text{ and }}x\in B\right\}}\,.$

3. Difference set

   ${}A\setminus B:={\left\{x\mid x\in A{\text{ and }}x\notin B\right\}}\,.$

For these operations to make sense, the sets need to be subsets of a common basic set. This ensures that we are talking about the same elements. Quite often, this basic set is not mentioned explicitly and has to be understood from the context. A special case of the difference set is the complement of a subset ${}A\subseteq G$ in a given set ${}G$, also denoted as

${}\complement A:=G\setminus A={\left\{x\in G\mid x\not \in A\right\}}\,.$

If two sets have an empty intersection, meaning ${}A\cap B=\emptyset$, we also say that they are disjoint.

Product set

We want to describe within set theory the arithmetic operations on the number sets mentioned above like addition and multiplication. For the addition (say on ${}\mathbb {N}$), two natural numbers ${}a$ and ${}b$ are added to yield another natural number, namely ${}a+b$. The set of pairs constitute the product set and the adding is interpreted as a mapping on the product set.

We define.

Definition

Suppose that two sets ${}L$ and ${}M$ are given. Then the set

${}L\times M={\left\{(x,y)\mid x\in L,\,y\in M\right\}}\,$

is called the product set

(or Cartesian product) of the sets.

The elements of a product set are called pairs and denoted by ${}(x,y)$. Here the ordering is essential. The product set consists of all pair combinations, where in the first component there is an element of the first set and in the second component there is an element of the second set. Two pairs are equal if and only if they are equal in both components.

If one of the sets is empty, then so is the product set. If both sets are finite, say the first with ${}n$ elements and the second with ${}k$ elements, then the product set has ${}n\cdot k$ elements. It is also possible to form the product set of more than two sets.

Example

Let ${}F$ be the set of all first names, and ${}L$ be the set of all last names. Then

$F\times L$

is the set of all names. The elements of this set are, in pair notation, ${}({\text{Heinz}},{\text{Miller}})$, ${}({\text{Petra}},{\text{Miller}})$ and ${}({\text{Lucy}},{\text{Sonnenschein}})$. From a name, one can easily deduce the first name and the last name by looking at the first or the second component. Even if all first names and all last names do really occur, not every combination of a first name and a last name does occur. For the product set, all possible combinations are allowed.

For a product set it is also possible that both sets are equal. Then one has to be careful not to confuse the components.

Example

A chess board (meaning the set of squares of a chess board where a chess piece may stand) is the product set

$\{a,b,c,d,e,f,g,h\}\times \{1,2,3,4,5,6,7,8\}.$

Every square is a pair, e.g., ${}(a,1),(d,4),(c,7)$. Because the two component sets are different, one may write instead of pair notation simply ${}a1,d4,c7$. This notation is the starting point to describe chess positions and complete chess games.

The product set ${}\mathbb {R} \times \mathbb {R}$ is thought of as a plane, one denotes it also by ${}\mathbb {R} ^{2}$. The product set ${}\mathbb {Z} \times \mathbb {Z}$ is a set of lattice points.

Example

The cylinder (its surface) is the product set of a circle and a line segment.

Let ${}S$ denote a circle (the circumference), and let ${}I$ be a line segment. The circle is a subset of a plane ${}E$, and the line segment is a subset of a line ${}G$, so that for the product sets, we have the relation

${}S\times I\subseteq E\times G\,.$

The product set ${}E\times G$ is the three-dimensional space, and the product set ${}S\times I$ is the surface of a cylinder.

Mappings

Suppose that a particle is moving in space. This process is described by assigning to every point in time ${}t\in \mathbb {R}$ the point in space ${}z(t)\in \mathbb {R} ^{3}$ where the particle is at that time. The running of a computer program that uses altogether ${}s$ memory units can be described by saying for every computing step (which means the execution of a program line) what the content of the memory units is. So to the number ${}n$ counting the computing steps, we assign the tuple

${}(b_{1}(n),\ldots ,b_{s}(n))\in \mathbb {N} ^{s}\,.$

In an election, every voter has to decide for exactly one party ( or for not voting). The temperature profile on the earth is described by assigning to every point in time and every point on the surface its temperature. Such and many other situations are captured by the concept of a mapping.

Definition

Let ${}L$ and ${}M$ denote sets. A mapping ${}F$ from ${}L$ to ${}M$ is given by assigning to every element of the set ${}L$ exactly one element of the set ${}M$. The unique element that is assigned to ${}x\in L$ is denoted by ${}F(x)$. For the mapping as a whole, we write

$F\colon L\longrightarrow M,x\longmapsto F(x).$

If a mapping ${}F\colon L\rightarrow M$ is given, then ${}L$ is called the domain (or domain of definition) of the map, and ${}M$ is called the codomain (or target range) of the map. For an element ${}x\in L$, the element

${}F(x)\in M\,$

is called the value of ${}F$ at the place (or argument) ${}x$.

Two mappings ${}F\colon L_{1}\rightarrow M_{1}$ and ${}G\colon L_{2}\rightarrow M_{2}$ are equal if and only if their domains coincide, their codomains coincide, and if for all ${}x\in L_{1}=L_{2}$ the equality ${}F(x)=G(x)$ in ${}M_{1}=M_{2}$ holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called functions. However, we will usually reserve the term function for mappings where the codomain is a number set like the real numbers ${}\mathbb {R}$.

For every set ${}L$, the mapping

$L\longrightarrow L,x\longmapsto x,$

which sends every element to itself, is called the identity (on ${}L$). We denote it by ${}\operatorname {Id} _{L}$. For another set ${}M$ and a fixed element ${}c\in M$, the mapping

$L\longrightarrow M,x\longmapsto c$

that sends every element ${}x\in L$ to the constant value ${}c$ is called the constant mapping (with value ${}c$). It is usually again denoted by ${}c$.

There are several ways to describe a mapping, like a value table, a bar chart, a pie chart, an arrow diagram, or the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule that allows computing the values of the mapping for every argument. Such rules are, e.g., (from ${}\mathbb {R}$ to ${}\mathbb {R}$) ${}x\mapsto x^{2}$, ${}x\mapsto x^{3}-e^{x}+\sin x$, etc. In the sciences and in sociology, also empirical functions are important that describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximately) in a mathematical manner.

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$
${}\pi (x)$	${}2$	${}4$	${}6$	${}5$	${}3$	${}1$

${}\cdot$	${}0$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$
${}0$	${}0$	${}0$	${}0$	${}0$	${}0$	${}0$	${}0$
${}1$	${}0$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$
${}2$	${}0$	${}2$	${}4$	${}6$	${}1$	${}3$	${}5$
${}3$	${}0$	${}3$	${}6$	${}2$	${}5$	${}1$	${}4$
${}4$	${}0$	${}4$	${}1$	${}5$	${}2$	${}6$	${}3$
${}5$	${}0$	${}5$	${}3$	${}1$	${}6$	${}4$	${}2$
${}6$	${}0$	${}6$	${}5$	${}4$	${}3$	${}2$	${}1$

Injective and surjective mappings

Definition

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called injective if for two different elements ${}x,x'\in L$, also ${}F(x)$ and ${}F(x')$

are different.

If we want to show that a certain mapping is injective then we may show the following: For any two elements ${}x$ and ${}x'$ fulfilling the condition ${}F(x)=F(x')$, we can deduce that ${}x=x'$. This is often easier to show than the statement that ${}x\neq x'$ implies ${}F(x)\neq F(x')$.

Definition

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called surjective if, for every ${}y\in M$, there exists at least one element ${}x\in L$, such that

${}F(x)=y\,.$

Example

We consider a football game as the mapping which assigns, to every goal of team ${}A$, the corresponding goal scorer. Suppose that there are no own goals and no changes. The goals of ${}A$ are numbered by ${}1,2,\ldots ,n$. Then we have a mapping

$\psi \colon \{1,\ldots ,n\}\longrightarrow A=\{{\text{player }}1,\,{\text{player }}2,\ldots ,{\text{player }}11\},$

given by

${}\psi (i)={\text{ the player who has scored the }}i{\text{-th goal}}\,.$

The injectivity of ${}\psi$ means that every player has scored at most one goal, the surjectivity means that every player has scored at least one goal.

Example

Let ${}H$ denote the set of all (living or dead) people. We study the mapping

$\varphi \colon H\longrightarrow H,$

which assigns to every person his or her (biological) mother. This is well-defined, as every person has a uniquely determined mother. This mapping is not injective, since there exists different people (brothers and sisters) with the same mother. It is also not surjective, since not every person is a mother of somebody.

Example

The mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is neither injective nor surjective. It is not injective because the different numbers ${}2$ and ${}-2$ are both sent to ${}4$. It is not surjective because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is injective, but not surjective. The injectivity can be seen as follows: If ${}x\neq y$, then one number is larger, say

${}x>y\geq 0\,.$

But then also ${}x^{2}>y^{2}$, and in particular ${}x^{2}\neq y^{2}$. The mapping

$\mathbb {R} \longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is not injective, but surjective, since every nonnegative real number has a square root. The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is injective and surjective.

Definition

Let ${}M$ and ${}L$ denote sets, and suppose that

$F\colon M\longrightarrow L,x\longmapsto F(x),$

is a mapping. Then ${}F$ is called bijective if ${}F$ is injective as well as

surjective.

Remark

The question, whether a mapping ${}F\colon L\rightarrow M$ has the properties of being injective or surjective, can be understood looking at the equation

${}F(x)=y\,$

(in the two variables ${}x$ and ${}y$). The surjectivity means that for every ${}y\in M$ there exists at least one solution

${}x\in L\,$

for this equation; the injectivity means that for every ${}y\in M$ there exists at most one solution ${}x\in L$ for this equation. The bijectivity means that for every ${}y\in M$ there exists exactly one solution ${}x\in L$ for this equation. Hence, surjectivity means the existence of solutions, and injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics, and they can also be interpreted as surjectivity or injectivity of suitable mappings.

Definition

Let ${}F\colon L\rightarrow M$ denote a bijective mapping. Then the mapping

$G\colon M\longrightarrow L$

that sends every element ${}y\in M$ to the uniquely determined element ${}x\in L$ with ${}F(x)=y$,

is called the inverse mapping of ${}F$.

Definition

Let ${}L,\,M$ and ${}N$ denote sets, let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

and

$G\colon M\longrightarrow N,y\longmapsto G(y),$

be mappings. Then the mapping

$G\circ F\colon L\longrightarrow N,x\longmapsto G(F(x)),$

is called the composition of the mappings

${}F$ and ${}G$.

So we have

${}(G\circ F)(x):=G(F(x))\,$

where the left-hand side is defined by the right-hand side. If both mappings are given by functional expressions, then the composition is realized by plugging in the first term into the variable of the second term (and to simplify the expression if possible).

Lemma

Let ${}L,M,N$ and ${}P$ be sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

$G\colon M\longrightarrow N,y\longmapsto G(y),$

and

$H\colon N\longrightarrow P,z\longmapsto H(z),$

be mappings. Then

${}H\circ (G\circ F)=(H\circ G)\circ F\,$

holds.

Proof

See Exercise 3.27 .

$\Box$

Footnotes

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