By a ‘set’ we understand any aggregation $\,M\,$ of certain well-differentiated objects $\,m\,$ of our view or thought (which are called the ’elements’ of $\,M\,$) into a whole.
What’s the point of sets?
Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, but we can give an informal description, describe important properties of sets, and give examples. All other notions of mathematics can be built up based on the notion of set.
Description
Similar (but informal) words: $\,$collection, group, aggregate.
Description: $\,$a set is a collection of objects which are called the members or elements of that set. If we have a set we say that some objects belong (or do not belong) to this set, are (or are not) in the set. We also say that sets consist of their elements.
If an object $m$ is an element of a set $M$, we write $m \in M$, read
“$m$ is an element of $M$”.If an object $m$ is not an element of a set $M$, we write $m \notin M$, read
“$m$ is not an element of $M$”.
Examples
- the set of students in this room
- the set of letters of the English language
- the set of natural numbers
- etc.
Sets can therefore consist of elements of various natures:
people, physical objects, numbers, signs, other sets, etc.
(We will use the words object or entity very broadly to include
all these different kinds of things.)
A set is an abstract object; its members do not have to be physically collected for them to form a set.
In principle, the membership criteria of a set must be well-defined, and not too vague. If we have a set and an object, it is possible that we do not know whether this object belongs to the set or not, because we do not have enough information or knowledge. (E.g. “The set of students in this room who are older than 18”: a well-defined set but we may not know who is in it). But the answer should exist, at least in principle. It might be unknown, but it should not be vague or fuzzy.
For example: Is the letter q the same as the letter Q? Well, that depends on which set we consider. If we take the set of 26 letters of the English alphabet, then q and Q are the same element. If we take the set of 52 upper and lower case letters of the English alphabet, then q and Q are two different elements. Both are possible, but we need to clarify which set we are talking about so that we know whether q = Q or not.
There is exactly one set, the empty set or null set, which has no members at all:
Impressive!
Specification of sets
There are two essential ways of specifying a set:
- by listing all elements of the set (list notation);
- by specifying a property of all its elements (predicate notation).
1. List notation
The list notation is mostly only suitable for finite sets. In this case, we enumerate the elements of a set, separate them with commas, and enclose them in curly braces.
Examples: $\{1, 3, 9\}$, $\{$red, green, blue$\}$, $\{a,b,d,m\}$
However, we do not always have to name all elements of a set, we can also skip elements, but it must always be clear which elements have been skipped. The Set $\{1,2,\,…,100\}$ contains all natural numbers from $1$ to $100$. We have clearly specified this set without listing all the elements of the set.
We can also specify non-finite sets with the ellipsis points
.... However, it must be clear and comprehensible how the listing would continue.clear continuation:
the natural numbers: $\{1,2,3,…\}$
the powers of two: $\{1,2,4,8,16,32,…\}$
the negative integers: $\{…,-3,-2,-1\}$unclear continuation:
$\{3,4,17,…\}$
$\{0,3,11,64,…\}$
Note that we don’t care about the order of elements and that elements can be listed multiple times. $\{1, 3, 9\}$, $\{9, 3, 1,3\}$ and $\{3,9, 3,1\}$ are different representations of the same set.
2. Predicate notation
Predicate notation defines a set of all objects satisfying a certain property $A(x)$. One writes $M = \{x\,|\,A(x)\}$ and means by this the set $M$ of all objects $x$, which satisfy the property $A(x)$.
Instead of a vertical bar, a colon is often used. That way $\{x\,:\,A(x)\}$ can also be written for the set $M$. The expression $\{x\,:\,A(x)\}$ is read as: “the set of all $\,x$ with $A(x)$” Some examples follow:
The set of all prime numbers :
$\{x : x$ is a prime number$\}$
The set of all real numbers between $-3$ and $5$ :
$\{y \in \R: -3 \lt y \lt 5\}$
The set of all odd natural numbers :
$\{z \in \N: \exists k \in \N: z = 2k-1\}$
Unlike the list notation, this set notation is also clear for non-finite sets. Therefore, for non-finite sets, this set notation should be used primarily.
Important sets
An overview of frequently used sets:
| Symbol | Notation | Description |
|---|---|---|
| $\N$ | $\{1,2,3, ...\}$ | natural numbers (without zero) |
| $\N_0$ | $\N \cup \{0\}$ | natural numbers with zero |
| $\Z$ | $\{...,-2,-1,0,1,2,...\}$ | integers |
| $\mathbb{Q}$ | $\{\frac{a}{b}\,|\, a \in \Z, b \in \N\}$ | rational numbers |
| $\mathbb{Q}^+$ | $\{\frac{a}{b}\,|\, a \in \N, b \in \N\}$ | positive rational numbers |
| $\R$ | real numbers | |
| $\R^+$ | $\{x \in \R\,|\, x \gt 0\}$ | positive real numbers |
| $\emptyset, \{\,\}$ | empty set |