Set

A set is an unordered collection of distinct objects. The objects in a set are typically referred to as its elements. A set is usually denoted by listing all of its elements between braces. For example, $\{1,3,2\}$ denotes a set of three numbers, and because sets are unordered, $\{3,2,1\}$ denotes the same set. $\{1,2,2,3,3,3\}$ does not denote a set, because the elements of a set must be distinct.

Another way to denote sets is the so-called abstraction method, in which the members of a set are given an explicit description, leaving no need for listing them. For example, the set from the example above $\{1,3,2\}$ can be described as the set of all natural numbers $x$ which are less than $4$. Formally that is denoted as $\{x\mid (x<4\left)\text{and}\right(x\text{is a natural number}\left)\right\}$.

Using the abstraction method allows for denoting sets with infinitely many elements, which would be impossible by listing them all. For example, the set $\{x\mid x=2n\text{for some natural}n\}$ is the set of all even numbers.

A set doesn't need to contain things of the same type, nor does it need to contain things that can all be brought to the same place: We could define a set $S$ that contains the apple nearest you, the left shoe which you wore last, the number 17, and London (though it's not clear why you'd want to). Rather, a set is an arbitrary boundary that we draw around some collection of objects, for our own purposes.

The use of sets is that we can manipulate representations of sets and study relationships between sets without concern for the actual objects in the sets. For example, we can say "there are 35 ways to choose three objects from a set of seven objects" regardless of whether the objects in the set are apples, people, or abstract concepts. ^[1]

It's also worth noting that a set of elements is itself a single distinct object, different from the things it contains, and in fact, one set can contain other sets among its elements. For example, $\{1,2,\{1,2\left\}\right\}$ is a set containing three elements: $1$, $2$ and $\{1,2\}$.

Examples

• $\{1,5,8,73\}$ is a set, containing numbers $1$, $5$, $8$ and $73$;
• $\left\{\right\{0,-3,8\left\}\right\}$ is a set, containing one element — the set $\{0,-3,8\}$;
• $\{\text{Mercury},\text{Venus},\text{Earth},\text{Mars}\}$ is a set of four planets;
• $\{x\mid x\text{is a human, born on 01.01.2000}\}$ is a set all people whose age is the same as the current year number;
• $\{\text{author's favorite mug},\text{Arbital's main page},73,\text{the tallest man born in London}\}$ is a set of four seemingly random objects.

Set membership

Main page: Set membership

Set membership can be stated using the symbols $\in$ and $\notin$. They describe the contents of a set. $\in$ indicates what is in a set. $\notin$ indicates what is not in a set. For example, "$x\in A$" translated into English is "$x$ is a member of the set $A$." and "$x\notin A$" translates to "$x$ is not in the set $A$."

Set cardinality

Main page: Cardinality

The size of a set is called its cardinality. If $A$ is a finite set then the cardinality of $A$, denoted $|A|$, is the number of elements $A$ contains. When $|A|=n$, we say that $A$ is a set of cardinality $n$. There exists a bijection from any finite set of cardinality $n$ to the set $\{0,...,(n-1\left)\right\}$ containing the first $n$ natural numbers. We can generalize this idea to infinite sets: we say that two infinite sets have the same cardinality if there exists a bijection between them. Any set in bijective correspondence with $N$ is called countably infinite, while any infinite set that is not in bijective correspondence with $N$ is call uncountably infinite. All countably infinite sets have the same cardinality, whereas there are multiple distinct uncountably infinite cardinalities.

See also

• Operations in Set theory - Common operations in set theory.
• Sets_examples - Examples of significant sets.
• Uncountability - A concrete exercise in comparing the cardinalities of infinite sets

1. ^{^︎}

   This is an example of abstracting over the objects.