**Sets**

A *set* is a collection or group of definable *elements* or *members*. Some examples of set elements are:

- points on a line
- instants in time
- individual apples in a basket
- coordinates in a plane
- coordinates in space
- coordinates on a display
- curves on a graph or display
- chemical elements
- individual people in a city
- 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.''

**Intersections and Unions**

**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 and only if

*x*

*A*and

*x*

*B*

The symbol is read ''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 and only if

*x*

*A*or

*x*

*B*

The symbol is read ''union.''

**Subsets**

A set *A* is a *subset* of a set *B*, written *A* *B*, if and 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 is read ''is a proper subset of.''

**Additional Sets**

**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 that

*x*

*B*

*x*

*B*implies that

*x*

*A*

Practice problems for these concepts can be found at:

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