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Definitions of Fundamental Concepts for Intermediate Algebra

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

A set is a collection of objects. The collection should be well-defined. That is, it must be clear that an object is either in the set or is not in the set. The objects in the set are called elements or members of the set. The members of a set may be listed or a description of the members may be given.

We list the members or describe the members within braces { }. Capital letters such as A, B, C, S, T, and U are employed to name sets. For example,

The symbol used to represent the phrase "is an element of" or "is a member of" is "." Thus we write 4 A to state that 4 is a member of set A. The symbol used to represent the phrase "is not an element of" is "." Hence 4 B is written to indicate that 4 is not an element of set B.

Sets are said to be equal if they contain the same elements. Hence {1, 5, 9} = {5, 9, 1}.

Note: Order is disregarded when the members of a set are listed.

Sometimes a set contains infinitely many elements. In that case it is impossible to list all of the elements. We simply list a sufficient number of elements to establish a pattern followed by a series of dots "…". For example, the set of numbers employed in counting is called the set of natural numbers or the set of counting numbers. We write N = {1, 2, 3, 4, …} to represent that infinite set. If zero is included with the set of natural numbers, the set of whole numbers is obtained. In this case the symbol used is W = {0, 1, 2, …}.

Set B above could be described as the set of multiples of three between 0 and 15. Note that the term "between" does not include the numbers 0 and 15. Set-builder notation is sometimes used to define sets. We write, for example,

B = {x | x is a multiple of three between 0 and 15}

There are occasions when a set contains no elements. This set is called the empty or null set. The symbol used to represent the empty set is "" or "{ }." Note that no braces are used when we represent the empty set by {negative natural numbers} is an example of an empty set.

Definition 1. Set A is a subset of set B if all elements of A are elements of B. We write A B

Hence, if A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6, 7}, A B. A is called a proper subset of B. If C = {4, 2, 6}, A C since the sets are the same set. A is called an improper subset of C.

New sets may be formed by performing operations on existing sets. The operations used are union and intersection.

Definition 2. The union of two sets A and B, written A B, is the set containing all of the elements in set A or B, or in both A and B.

If set-builder notation is used, we write A B = {x | x A or x B}. If set A = {1, 5, x, z} and B = {3, 5, 7, z}, then A B = {1, 5, x, z, 3, 7}. Recall that the members of a set may be listed without regard to order.

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