Working with Sets

A set is a well-defined collection of distinct objects. The objects of a set are called elements. A set is well-defined, if it can be easily understood, whether, a given object, is an element of that set. Some sets do not have any elements. These sets are called empty sets, or null sets. They are denoted by using the symbol Ø.

N = Ø

A set of digits could be defined as:

D = {1, 2, 3, 4}

The set is labeled D, and includes the digits 1, through 4. The left: {, and right: }, braces, are used to enclose all the elements of the set. This makes it easy to understand what is, and is not, an element of the set D. This method of defining a set is called the roster method.

Set-builder Notation is another method that can be used to define a set. An example of the same set, using set-builder notation would be:

D = {x|x is a digit between 1, and 4, inclusive}

This set would be read as “D is the set of all x, such that, x is a digit between 1, and 4, inclusive.” The horizontal pipe: |, is pronounced “such that.” Therefore, when “x|x”, is used, it should be read as “x, such that, x.”

When the word “inclusive”, is used, it means that both 1, and 4, are also included in the set; otherwise, a digit between 1, and 4, would be either 2, or 3. This is because 2, and 3, are the only digits that are between 1, and 4.

Z = {x|x is a digit between 1, and 4}

In this case, set D, would have 4 elements, and set Z, would have 2 elements.

Elements Are Unique

Elements in a set should be unique, i.e., they should not appear more than once in the set. You would not create a set such as: {3, 9, 1, 3}, because the element “3”, appears more than once in the set. The order of the set does not matter. Both: {3, 6, 9} and: {6, 9, 3} represent the same set.

Equal Sets

When two sets are the same, such as: X = {5, 4, 3}, and Y = {3, 4, 5}, then it can also be said that X = Y. The set of both X, and Y, are the same, even though the elements are not displayed in the same order in both sets.

Subsets

If one set contains all the elements of another set, it is called a subset. As an example: P = {4, 9, 0}, and: Q = {0, 9, 6, 4, 3}. In this case, P, is a subset of Q, because every element in P, is also an element in Q. The set of Q also contains 4, 9, and 0. When a set is a subset of another set, it is written as: P ⊆ Q.

Intersection

The intersection of a set is defined as: A ∩ B. If a problem uses this format, it’s simply asking for a list of all elements, that appear in both set A, and, set B. So, if the element “5”, appears in both sets, and all other elements are unique, then the answer would be {5}.

A = {4, 5, 9}; B = {5, 1, 2}; Then A ∩ B = {5}.

Union

The union of a set is defined as: A ∪ B. If a problem uses this format, it’s simply asking for a list of all elements, that appear in either A, or B, or both. If an element appears in both sets, it is only listed once in the union. Look at element “8” in the sample:

A = {9, 8, 4}; B = {2, 1, 8}; Then A ∪ B = {9, 8, 4, 2, 1}.

Universal Set

A universal set, written as ∪, is a set that consists of all elements that need to be considered.

Set Complement

The complement of a set, written as Ā (with a line above the set name), is simply a list of all of the elements in the ∪ (universal) set, that are not in set A. In this case, if ∪ = {1, 2, 3, 4, 5, 6, 7, 8, 9}, and A = {6, 7, 9}, then Ā = {1, 2, 3, 4, 5, 8}.

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