Quantities
Kapitoly: Quantities, Plural operations, Countable sets, Paradoxes of set theory
A set can be thought of as a collection of elements. Thus, every set contains a certain number of elements, which can be finite or infinite. It may also not contain any element, then we speak of an empty set. We usually denote a set by a capital letter, for example M, and the elements of a set by a lowercase letter m.
What is a set
A set is one of the basic concepts in mathematics, through which a bunch of other things are defined, so it's a good idea to be clear at the start what a set is so that you don't get confused later on in your learning.
So a set is some collection of arbitrary elements. In mathematics, we most often work with numerical sets, that is, sets whose elements are numbers. The classical notation of a set in mathematics is as follows:
$$M=\left\{1{,}2,3\right\}$$
We have thus described a set called "M", which contains three elements, one, two and three. We always write sets in compound brackets; practically any time you see compound brackets, it is a set of some sort.
What can we use sets for? Sets often tell us what elements we can choose. For example, in everyday life, you might say something like "Agnes, let's play a game, shall we? Think of a number from one to five." You've given Agnes a range of numbers to choose from. In maths, you'd use a set:
$$L=\left\{1{,}2,3{,}4,5\right\}$$
And then you could say to Agnes, "Agnes, let's play a game, okay? Think of a number that is contained in the set L." I'm sure the game would be more fun. In mathematics, we use ∈ to write that an element belongs to a set, and if it doesn't, we use $ \notin$. So if we want to say that one belongs to the set L, but seven doesn't, we would write it like this: 1 ∈ L and $7 \notin L$.
Of course, the set can be empty, this is done either by writing P = {} or more simply P = ∅. Note that both entries mean "empty set", if you wrote it like this: P = {∅}, you would write a set that contains an empty set in it. It is not the same as an empty set.
Disorder and duplication
A set is not said to have elements that are ordered in some way. A set contains a disordered set of elements. If we had two sets A = {1, 2, 3} and B = {3, 2, 1} we would say they are the same. The order of the elements in the set simply does not matter.
Nor do we care about duplicate elements. If a set contains multiple identical elements (multiple identical numbers), we only ever consider a single occurrence of that element. Again, if we had these two sets A = {1, 1, 2, 2, 2} and B = {2, 1} we would consider them the same. It doesn't matter that the set A contains "more" elements because it contains duplicated or triplicated elements. When counting with sets, we simply filter out these duplicated elements.
Size and equality
We can define the notion of set size, which is the number of elements in a set. Thus, from the previous example of A = {1, 1, 2, 2, 2} and B = {2, 1}, the size of the set A would be two, but the size of the set B is also two, because we are not interested in duplicate elements when counting the elements of a set either. We denote the size of the set by vertical lines: |A| = |B| = 2.
As you may have already understood, two sets are equal if both sets have the same elements. Some examples:
$$\begin{eqnarray} \left\{1, 2, 3\right\}&=&\left\{1, 2, 3\right\}\\ \left\{1, 2, 3\right\}&=&\left\{1, 2, 3, 2, 3, 1\right\}\\ \left\{a, h, o, j\right\}&=&\left\{o, o, h, j, a, o\right\}\\ \left\{1, 3, 5, 9\right\}&\ne&\left\{1, 3, 9\right\}\\ \emptyset&\ne&\left\{x\right\} \end{eqnarray}$$
An essential property is that sets can contain as their element a set again. Example: C = {1, 2, {3, 4, 5, 6}}. It is important to note that the set C is a three-element set, not a six-element set. The set C contains three elements: a one, a two, and a set. The elements 3, 4, 5 and 6 are contained in the inner set, not the set C. So |C| = 3 applies. A more complicated example:
$$D=\left\{0, \left\{1, \left\{2, 3\right\}\right\}, \left\{4\right\}\right\}$$
How many elements does the set D contain ? It contains three elements, these are the elements:
$$D_1=0,\qquad D_2=\left\{1, \left\{2{,}3\right\}\right\},\qquad D_3=\left\{4\right\}$$
The set can be finite or infinite. Finite sets are all the ones we have mentioned so far. For example, the set of all numbers is infinite.
A subset of
Consider two sets A = {1, 2} and B = {1, 2, 3}. these sets are different because they do not contain the same elements, the set B is larger. However, you will have noticed that the set B contains exactly the same elements as the set A, only it has the extra element 3. At this point we can say that A is a subset of B.
If A is a subset of B, then it must be true that all the elements contained in the set A must also be contained in the set B. Being a subset is a relation and we write it using the symbol ⊆. Formal definition:
$$A \subseteq B \Leftrightarrow \forall x \in A:\quad x \in B$$
It is commonly assumed that the sets A and B can be the same and A ⊆ B will still hold. If we want to express a sharp variant of a subset, we use a different symbol: ⊂. Then if A ⊂ B, then the set B must be larger (if finite), it must contain an element that the set A does not contain. We then call such a set a "proper subset". Thus, if A ⊂ B, then A is a proper subset of B. Definition of proper subset:
$$A \subset B \Leftrightarrow (A\subseteq B \quad\wedge\quad A \ne B)$$
The definition is the same as for the classical subset, except that the two sets must not be equal. Some examples:
$$\begin{eqnarray} \left\{a, h, o\right\}&\subseteq&\left\{a, h, o, j\right\}\\ \left\{a, h, o\right\}&\subset&\left\{a, h, o, j\right\}\\ \left\{2, 4, 6\right\}&\subseteq&\left\{2, 4, 6, 8, \ldots\right\}\\ \left\{2, 4, 6\right\}&\subset&\left\{2, 4, 6, 8, \ldots\right\}\\ \left\{1, 2, 3\right\}&\not\subseteq&\left\{1, 3\right\}\\ \left\{1, 2, 3\right\}&\not\subset&\left\{1, 3\right\}\\ \left\{0, 1\right\}&\subseteq&\left\{0, 1\right\}\\ \left\{0, 1\right\}&\not\subset&\left\{0, 1\right\}\\ \emptyset&\subseteq&\left\{\pi\right\}\\ \emptyset&\subset&\left\{\pi\right\}\\ \emptyset&\subseteq&\left\{\emptyset\right\}\\ \emptyset&\subset&\left\{\emptyset\right\}\\ \left\{0\right\}&\not\subseteq&\emptyset\\ \left\{0\right\}&\not\subset&\emptyset\\ \left\{\diamond, \bigtriangleup, \odot, \ddagger, \wr\right\}&\subseteq&\left\{\diamond, \bigtriangleup, \odot, \ddagger, \wr, \star, \bullet, \mp\right\}\\ \left\{\diamond, \bigtriangleup, \odot, \ddagger, \wr\right\}&\subset&\left\{\diamond, \bigtriangleup, \odot, \ddagger, \wr, \star, \bullet, \mp\right\} \end{eqnarray}$$
Properties of a subset:
- A ⊆ A: a set is always its own subset.
- A ⊄ A: a set is never a subset of itself.
- ∅ ⊆ A: an empty set is a subset of any set.
- A ⊄ ∅: an empty set has no proper subset.
We can use subsets to write the equality of sets:
$$A = B \quad\Leftrightarrow\quad A \subseteq B \wedge B \subseteq A$$
If both sets are equal, then one is a subset of the other.
We can also denote a subset by the word "inclusion".
A potential set
A potential set is the set of all subsets of a given set. It is usually denoted by either P(M) or 2M.
Example: M = {1, 2, 3}. What are all the subsets? Certainly the empty subset and the set itself is M. Next: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}. All of these sets form the power set of the set M. Write: P(M) = {∅, M, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}.
Since the empty set is a subset of every set and M is always a subset of M, it will always be true that ∅ ∈ P(M) and M ∈ P(M).
How to write a set
We have already mentioned one way, by simply listing the elements. We use compound parentheses to do this. M = {1, 2, 3} Or N = {a, b, c, d} etc. If we are writing an infinite set, we can use three dots to do this, as long as it is obvious how the sequence of elements will continue: M = {1, 2, 3, …}
The main way to write a set is the characteristic property of the set. In general, the notation would look like this: {x ∈ X | P(x)}, where X is the set from which we select elements and P(x) is some formula that specifies the elements of the set. The formula can be written purely mathematically or verbally. A semicolon is also used instead of "|": ";".
For example, "let the set M contain all the numbers that denote some day of the month". The month has at most 31 days, so such a set would have 31 elements, from 1 to 31: M = {1, 2, 3, …, 30, 31}. Another notation for the same set might look like this: $\left\{x \in \mathbb{Z} | \mbox{ x denotes the day of the month }\right\}$, where ℤ denotes the set of integers.
A more mathematical example might be "let the set P contain all positive numbers that are divisible by five". Then the set would look like this: P = {5, 10, 15, 20, 25,…}
Let's try to write down mathematically the set T of natural numbers that are less than ten: T = {x ∈ ℕ | x<10}. This notation tells us: the set T consists of the elements x, which we take from the set of natural numbers and which satisfy the condition that they are less than 10. So we look at the natural numbers and return only those that are less than ten: {1, 2, 3, 4, 5, 6, 7, 8, 9}. And that's it.
Another example: Y = {x ∈ ℤ | x ≠ 0}. We have defined a set Y, which contains the elements x, which we take from the integers, and the only condition that x must satisfy is that it must not be zero. Thus the set Y contains all integers except zero.
Another example: G = {x ∈ ℝ | x · x = x}. The set G contains the elements of x, which we take from the real numbers, and for all elements of x it must hold that if we multiply them with themselves we get again the number x. For example, we can try the number 5. By definition, the following equality should hold: 5 · 5 = 5. Obviously this does not hold, so the number 5 will not be an element of the set G. Let's try one: 1 · 1 = 1. Obviously this is true for it, so one will be an element of the set G. The second, and last, element will be the number zero. This will not be true for any others. So we can write: G = {0, 1}.
One last example. I'll write something more complicated so you can see that a characteristic property can be complex:
$$X=\left\{x \in \mathbb{R} | (x^2=2x)\vee(sin(x)=\pi\wedge cos(x)=\pi)\right\}$$
A set is in practice most often defined by a characteristic property, and it can be said that a pretty large fraction of concepts in mathematics are defined by sets. Taking such an interval, we can say that the interval (a, b) is the set I, for which:
$$I=\left\{x\in\mathbb{R} | (x > a) \wedge (x < b)\right\}$$
These are all elements of the set of real numbers that are greater than a and less than b, which is exactly what the interval expresses.