# Sets

A set is a collection of **elements**. It is described by all its elements.

The set $$\N$$ of the natural numbers consists of 1, 2, etc.

The set of the letters of ALI BABA consists of the elements A, L, I and B.

When we define the set by listing its elements, we put them in **braces** (curly brackets { }) **separated by commas**.

The set of the letters of ALI BABA is written as $$\{A,B,I,L\}$$.

The set of the integers can be written as $$$\Z =\{0, 1, -1, 2, -2, 3, -3, \dots\}$$$

The **order** in which the elements are written **does not matter**.

$$\{0,1,3,5\}$$ and $$\{3,1,5,0\}$$ are the same set.

We say that an element belongs to a set if it is part of the set. We use the symbol $$\in$$.

$$$ 10\in\Z,\;L\in\{A,B,I,\Tred{L}\} $$$ When an element does **not** belong to a set we use $$\notin$$.

A set $$A$$ is a subset of $$B$$ if all the elements of $$A$$ belong to $$B$$. We write this $$A\subseteq B$$.

The set of mammals is a subset of the set of all animals.

The set of the natural numbers $$\N$$ is a subset of the integers $$\Z$$. But the set $$\Q$$ of the irrational numbers is not.

The set of vowels is a subset of the set of all letters.

$$A$$ is a proper subset if it is a subset different from $$B$$. It contains some elements of $$B$$ but not all of them. We write this as $$A\subset B$$.

$$$ \N\subset\Z,\;\Z\subseteq\Z,\;\{\Tred{A},\Tred{I}\}\subset\{\Tred{A},B,\Tred{I},L\} $$$The subset of a subset is a subset.

$$$ \N\subset\Z\subset\Q\subset\R$$$ $$A$$ is **not a subset** of $$B$$ if $$A$$ contains an element that is not in $$B$$. $$A$$ and $$B$$ can still have elements in common. We write this $$A\not\subset B$$.

The set of animals that live in water is not a subset of all fish (whales are mammals).

$$$ \Z\not\subseteq\N,\quad\Z\not\subset\Z,\quad \{\Tgreen{A}, \Tred{O}, \Tgreen{I}\}\not\subset\{\Tgreen{A}, B, \Tgreen{I}, L\}$$$ The empty set has **no** elements. We write it $$\emptyset$$. It is a subset of **all sets**.

The set of all men with a height of 5m or more is an empty set.

A finite set is a set with a **finite** (limited) **number of elements**.

The set of the letters in the English alphabet is finite. There are 26 elements (the letters) in the alphabet.

The set of the natural numbers smaller than 10 is finite.

The set of all human beings is finite.

If the **number of elements** in a set is **infinite**, we have an infinite set.

The set of the natural numbers is infinite.

The set of the integers smaller than 10 is infinite: it consists of 9, 8, ..., 0, -1, -2, ...

The set of all possible words (whether they mean anything or not) is infinite.

The number of elements of a **finite** set $$A$$ is written as $$n(A)$$ or $$\vert A \vert$$. It is also called the cardinality of the set $$A$$.

A **subset** $$A$$ of a finite set $$B$$ is finite and has fewer elements than the set $$B$$. This can be written as $$$n(A)\le n(B).$$$

The complement of a subset $$A$$ of $$E$$ consists of all the elements in $$E$$ that are **not** in $$A$$. It is written as $$A'$$ or $$A^c$$.

The complement of the set of the odd numbers is the set of even numbers (with the set of integers as the reference set).

The complement of the set of the vowels is the set of the consonants (with the set of English letters as the reference set).

The reference set $$E$$ is often omitted when it is clear which set we are referring to.

The **complement of the complement** of a set is the original set $$$ (A')' = A.$$$

The union of two sets $$A$$ and $$B$$ is a set that consists of the elements that are **either** in $$A$$ or $$B$$ or in ** both**. The set is written $$A\cup B$$.

The union of the odd and even numbers is the set of all the integers.

$$$ \{\Tblue{1},\Tblue{2},\Tblue{3}\}\cup \{\Tgreen{0},\Tgreen{2},\Tgreen{4}\} = \{\Tgreen{0},\Tblue{1},\Tred{2},\Tblue{3},\Tgreen{4}\}$$$

$$A$$ and $$B$$ are a subset of $$A\cup B$$.

The intersection of two sets $$A$$ and $$B$$ consists of the elements that are in **both** $$A$$ and $$B$$. The set is written $$A\cap B$$.

$$$ \{1,\Tred{2},3\}\cap \{0,\Tred{2},4\} = \{\Tred{2}\}$$$

The intersection of the odd and even numbers is empty.

The intersection of the multiples of $$2$$ and of the multiples of $$3$$ is the set of the multiples of $$6$$.

The intersection $$A\cap B$$ is a subset of both $$A$$ and $$B$$.

Two sets are mutually exclusive, or disjoint, if they have no elements in common. Their intersection is empty.

The set of the people who lived in the 15th century and the set of the people who live today are mutually exclusive.

Here is a summary of the main set operations with the following subsets $$A$$ and $$B$$ of $$E$$ $$$ A = \{1,2\},\quad B=\{2,3\},\quad E=\{0,1,2,3\}.$$$

Operation | Complement | Union | Intersection |
---|---|---|---|

Notation | $$A'$$ | $$A\cup B$$ | $$A\cap B$$ |

Meaning | Not in | In either or both | In both |

Example | $$\{0,3\}$$ | $$\{1,2,3\}$$ | $$\{2\}$$ |

Venn Diagram |