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Set theory

Set theory is about how elements can be divided into groups or sets.

An element can be anything.

For example:

We have three elements: a banana, an orange and a pear.

These could be described by the fact that, they belong to the set of fruits.

The concepts of set theory

Here is an example of two sets called M and N.
Two sets with an intersection

The numbers 1, 2, 3, 4 and 5 are members of M, and the numbers 4, 5, 6 and 7 are members of N.

The two sets are denoted like this:

M={1,2,3,4,5} and N={4,5,6,7}

Sets are usually named with capital letters.

Set membership

The number 2 is a member of the set M. This can be denoted with the “element of”-symbol \in.

\{1\}\inM

The number 7 is not a member of the set M This can be denoted with the “not element of”-symbol \notin:

\{1\}\notinM

The empty set

The empty set is a set having no elements. This is denoted by Ø or by curly brackets with nothing inside {}.

Union

The union of sets is all the elements from all sets. The sets are added together.

M U N=\{1,2,3,4,5\} U \{4,5,6,7\}=\{1,2,3,4,5,6,7\}

Intersection

The intersection of sets is the elements, which the sets have in common.

M n N=\{1,2,3,4,5\} n \{4,5,6,7\}=\{4,5\}

Relative complement

The relative complement of sets is the elements, which are members of the one set but not the other. The other set is subtracted from the one set.

M\backslashN=\{1,2,3,4,5\}\backslash\{4,5,6,7\}=\{1,2,3\}

Compliment

The compliment of a set is the elements, which are not members of that set.

Given is the two sets:

M={1,2,3,4,5} and N={4,5,6,7}

The complement of the set M is the elements not being member of M:

MC={6,7}

Subset

Example of a subset of a set

Given is the two sets:

B={1,2,3,4} and A={3,4}

As you can see, all elements belonging to the set A also belong to the set B, which makes A a subset of B.

This is denoted by:

A\subseteqB

This also means that A has the same or fewer elements than B.
B is not a subset of A, because there are elements in B, which are not members of A.

Proper subset

Given is the two sets:

B={1,2,3,4} and A={3,4}

A is the subset of B, but A is also the proper subset of B, because B has elements, which are not members of A. So A is smaller than B.

This is denoted:

A\subsetB

Full “set-builder notation”

Subsets of sets can be expressed by using the “such that”-symbol, which is a vertical bar.

Given is the set:

A={1,2,3,4}

Now we want to make a subset of A with all numbers greater than 2:

\{x\inA|x>2\}=\{3,4\}

This is pronounces as: “x is an element of A, such that x is greater than 2”. The elements of this subset are the numbers 3 and 4.