Element

An element is a distinct entity which is a member of a set. This means that an element is anything that can be named, measured, and contained within a set.

Set

A set describes a collection of elements, or distinct independent entities. Grouping elements together in sets is how we define relationships between otherwise unrelated entities.

Element of

A symbol which correlates an element to a set it belongs to. This may be read: "the element x is a member of set A." An element can belong to multiple sets; but, a set may only contain one of each element.

Not Element of

A symbol which represents the exclusion (or the disconnection) of an element from a set. This may read: "the element x is not a member of set A."

Equality

Equality is how we express similarity, or likeness between distinct entities. This can be equal: quantities, sets, expressions, or anything else that can be described as an element.

Equal Sets

Set A is equal to set B iff (if and only if) all of the members of set A match those of set B (without any extra or excluded elements).

Not Equals

Set A does not equal set B if any of the members contained in either set do not match a member contained in the other set.

Empty Set/Null Set

Describes an empty set, or a set with no members.

Subset (Not Equals)

Set A is a subset of set B iff set B contains every element contained within set A and at least one more element in which set A does not.

Subset Equals

As with the plain subset, A is a subset of B iff every element of A is a member of set B; however, contrary to the plain subset, set A may equal set B.

Cardinality

The size of a set, or the number of members contained within a set.

Summation

Represents the action of adding together the elements of set A with those of set B in a distributive fashion. This is done by taking each member from set A and summing it with every member in set B, with each sum being a member in the resulting set.

Universal Set

The universal set is the set which contains all elements considered during set operations. This does not only contain elements which are a member of a designated set but all possible elements that may be a member of a given set.

Complementation

The complement of a set is its opposite, or all of the elements the set does not have. The complement of set A is the universal set U minus set A: A'=U-A.

Set A

Set B

Union

Intersection

Symmetric Difference

Relative Complement

Real Numbers

A set which has every real number in existence as its members. Entities that are members of this set are called 'real numbers,' and it is with these elements that mathematics is formed.

Irrational Numbers

A set which includes all numbers which are not rational. These numbers are called 'irrational' and they include numbers with an infinitely non-repeating fractional part or with an imaginary component (i).

Rational Numbers

Any number that can be broken down into a ratio of two integers is called 'rational'. This includes all numbers with terminating, or repeating fractional components.

Integers

A set containing all the real numbers (positive and negative) without fractional components.

Whole Numbers

A set containing all integers that are greater than or equal to zero.

Natural Numbers

A set containing all integers that are greater than but not equal to zero.

Complex Numbers

A set containing all numbers with an imaginary component (i).