# Sets – Intersection, Union, Subsets, Disjoints

A set is a collection or group of definable elements or members. Set elements commonly include:

- points on a line
- instants in time
- coordinates in a plane
- coordinates in space
- coordinates on a display
- curves on a graph or display
- physical objects
- chemical elements
- locations in memory or storage
- data bits, bytes, or characters
- subscribers to a network

If an object or number (call it a) is an element of set A, this fact is written as:

a ∈ A

The ∈ symbol means “is an element of” or “is in”.

**Set Intersection**

The intersection of two sets A and B, written A ∩ B, is the set C such that the following statement is true for every element x:

x ∈ C if an only if x ∈ A and x ∈ B

The ∩ symbol means “intersect”.

**Set Union**

The union of two sets A and B, written A ∪ B, is the set C such that the following statement is true for every element x:

x ∈ C if an only if x ∈ A or x ∈ B

The ∪ symbol means “union”.

**Subsets**

A set A is a subset of a set B, written A ⊆ B, if an only if the following holds true:

x ∈ A implies that x ∈ B

The ⊆ symbol is read “is a subset of.” In this context, “implies that” is meant in the strongest possible sense. The statement “This implies that” is equivalent to “If this is true, then that is always true.”

**Proper Subsets**

A set A is a proper subset of a set B, written A ⊂ B, if and only if the following both hold true:

x ∈ A implies that x ∈ B

as long as A ≠ B

The ⊂ symbol means “is a proper subset of”.

**Disjoint Sets**

Two sets A and B are disjoint if and only if all three of the following conditions are met:

A ≠ ∅

B ≠ ∅

A ∩ B ≠ ∅

where ∅ denotes the empty set, also called the null set. It is a set that doesn’t contain any elements, like a basket of apples without the apples.

**Coincident Sets**

Two non-empty sets A and B are coincident if and only if, for all elements x, both of the following are true:

x ∈ A implies x ∈ B

x ∈ B implies x ∈ A