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A set X is an unordered collection of objects, which are called the elements of X. For example, the three languages French, Russian and Chinese form a set, which we denote as X = {French, Russian, Chinese} (or equivalently, {Chinese, French, Russian} as order does not matter).

Subset (or cluster)

A set Y is a subset of a set X if every element in Y is also contained in X. For example, {French, Russian} is a subset of the set {French, Russian, Chinese}. Sometimes a subset of a set which contains at least one element is also known as a cluster.

Set union

The union of two sets X and Y, denoted X U Y is the set formed by taking all of the elements in X and all elements in Y. For example, {French, German} U {Italian, Russian, English} = {French, German, Italian, Russian, English}

Set intersection

The intersection of two sets X and Y, denoted XY, is the set formed by taking all those elements that are in both X and Y. For example, {French, German} ∩ {Italian, Russian, French} = {French}.

Empty set

The empty set is the set that contains no elements, which is usually denoted Ø.

Disjoint sets

Two sets are called disjoint if their intersection is the empty set, that is, they have no elements in common. So, for example, the two sets {French, German} and {Italian, Russian, English} are disjoint sets as their intersection is the empty set, i.e. {French, German} ∩ {Italian, Russian, English} = Ø.

Complement of a set

If Y is a subset of a set X then the complement of Y in X, denoted by X-Y, comprises of all those elements in X that are not in Y. For example, the complement of {French, Chinese} in the set {French, Chinese, Russian, Italian} is {Russian, Italian}.