# Sets: Terminologies, Notations, and Operations

As a preparation for more posts on probability, statistics, permutations and combinations, we familiarized ourselves last week with the different terminologies and notations of probability. We continue in this post by studying set terminologies, notations, and operations. Note that this is also the third post in the **Set Primer Series;** the first and second are Introduction to Sets and Subset: a set contained in a set.

**Universal Set**

The *universal set* is the set that contains all the elements under discussion. If we talk about the letters in the English alphabet, then the universal set contains all the 26 letters. In set theory, universal set is usually denoted by .

In the following discussion, we let be the set of integers, be the set of even integers, and be the set of odd integers. The following are the common operations on sets.

**Intersection**

If sets and have elements in common they form a set written as . This is the *intersection* of and .

Example: If we let and then .

**Union**

The *union* of sets , , is the set containing all the elements of and or both.

Example: If we let and then .

Example: If we let be the set of even integers and be the set of odd integers, then .

**Disjoint**

If and have no elements in common, they are said to be *disjoint*.

Example: The sets and mentioned above have no elements in common so they are disjoint sets. In notation, we use the symbol to denote an empty set. Therefore, .

Example: From above, it is also clear that

**Difference**

is the set consisting of all elements of which are not in .

Example: .

**Complement**

The set of elements in which are not in is the complement of denoted by .

Example: If we let be the set of integers, be the set of odd integers, then .

Image Source: Wikipedia