Geometry Unit 1 Lesson 5 More on Grouping

A collection of well-defined objects, called elements, which is described either by a listing of all the objects (a roster), or by a symbolic description that specifies the object in a set (a rule). In mathematics, we use braces { } to show a set, and name it using a capital, or upper-case letter.

One of the objects in a set. We indicate thsi relationship by using a lower-case Greek letter "epsilon" (ϵ), as follows: "2 is an element of set A" would be represented by 2 ϵ A

A set containing a quantity of members, or elements, which cannot be counted. An ellipsis (...) is used in such a set, to indicate that quality.

A set whose members can be counted, because it has a definite number of elements.

A set which contains no members. it is sometimes called the "null" set and is denoted by empty braces { }. (or ∅) A null set is also a subset of every set.

A mathematical shorthand used to describe a set. FOr example, "the set of all numbers represented by n, such that n is a multiple of 5" would be represented by {n | n is a multiple of 5}

An operation on two or more sets, which unites or combines the sets, so that all of the elements in all of the sets are listed in a single set. Each element appears only once in the union, even if it appears in both sets. This operation is symbolized by a "cup" (∪). Fore example, "the union of sets X adn Y" would be represented by X ∪ Y.

An operation on two or more sets, which selects only those elements common to (or belonging to) all of the original sets. This operation is symbolized by a "cap" (∩). For example, "the union of sets X and Y" would be represented by X ∩ Y.

Two or more sets which have no members, or elements, in common.

Two or more sets which contain exactly the same elements. This relation is symbolized by the standard equality symbol (=). Fore example, "Set M is equal to set N" would be represented by M = N.

A set, all of whose elements are contained in another set. This relation is symbolized as follows: "Set E is a subset of set F" would be represented by E ⊆ F. Notice that this symbol is similar to the numberical relation symbol ≤, meaning "is less than or equal to" and is consistent with the concept of a subset containing either part of all of another set.

A subset which contains only part of another set. This relation is symbolized as follows: "Sek K is a proper subset of set L", would be represented by K ⊂ L. Notice that this symbol is similar the the numerical relation symbol <, meaning "is strictly less than" and is consistent witht he concept of a proper subset containing fewer elements than another set.

A subset which contains all of the elements of another set. In other words, the subset is equal to the original set.

A pictoral representation of the relationships betwen sets, within a Universal set. The Universal set is shown by a rectangle and is represented by U. All other sets are shown by circles, placed appropriately within the rectangle.