Set Theory Basics: Learn It 1

  • Understand the concepts of sets, including empty sets, subsets, and proper subsets, and use correct set notation
  • Describe and perform set operations (union, intersection, complement, and difference) using proper set notation
  • Create and interpret Venn diagrams to represent and analyze set relationships and operations
  • Apply the concepts of sets, subsets, and cardinality properties to solve real-life problems

A movie lover might own a collection of movie posters, while a music lover might keep a collection of vinyl records. Any collection of items can form a set.

set

A set is a collection of distinct objects, called elements of the set.

 

A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets separated by commas.

 

Repeated elements in a set are only listed once. A set simply specifies the contents; order is not important.

 

The set represented by [latex]\{1, 2, 3\}[/latex] is equivalent to the set [latex]\{3, 1, 2\}[/latex].
Some examples of sets defined by describing the contents:

  1. The set of all even numbers
  2. The set of all books written about travel to Chile

Some examples of sets defined by listing the elements of the set:

  1. [latex]\{1, 3, 9, 12\}[/latex]
  2. [latex]\{\text{red, orange, yellow, green, blue, indigo, purple}\}[/latex]

set notation

Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.

 

The symbol [latex]\in[/latex] means “is an element of”.

 

A set that contains no elements, [latex]\{ \}[/latex], is called the empty set or null set and is notated [latex]\emptyset[/latex].

Let [latex]A=\{1,2,3,4\}[/latex]. To notate that [latex]2[/latex] is element of the set, we’d write [latex]2 \in A[/latex].

Sometimes a collection might not contain all the elements of a set. For example, Marta owns ninety-five Pokémon cards. While Marta’s collection is a set, we can also say it is a subset of the larger set of all Pokémon cards.

subset

A subset of a set [latex]A[/latex] is a set that consists solely of elements drawn from set [latex]A[/latex]. It may include all, some, or none of set [latex]A[/latex]‘s elements. In other words, every element in the subset is also an element of set [latex]A[/latex].

 

If [latex]B[/latex] is a subset of [latex]A[/latex], we write [latex]B \subseteq A[/latex]

 

A proper subset of a set [latex]A[/latex] is a subset that contains some, but not all, of the elements of set [latex]A[/latex]. It is a subset that is not identical to the original set, meaning it must contain fewer elements than set [latex]A[/latex].

 

If [latex]B[/latex] is a proper subset of [latex]A[/latex], we write [latex]B \subset A[/latex]
Given the set: [latex]A = \{a, b, c, d\}[/latex]. List all of the subsets of [latex]A[/latex].

Listing the sets is fine if you have only a few elements. However, if we were to list all of the subsets of a set containing many elements, it would be quite tedious. Instead, in the next example we will consider each element of the set separately.

Given the set: [latex]A = \{a, b, c, d\}[/latex], there are four elements.

For the first element, [latex]a[/latex], either it’s in the set or it’s not. Thus there are [latex]2[/latex] choices for that first element.

Similarly, there are two choices for [latex]b[/latex] — either it’s in the set or it’s not.

Using just those two elements, list all the possible subsets of the set [latex]\{a,b\}[/latex].

Exponential Notation

Recall that the expression [latex]a^{n}[/latex] states that some real number [latex]a[/latex] is to be used as a factor [latex]n[/latex] times.

Ex. [latex]2^{5} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32[/latex]

You may be asked to find the number of subsets and proper subsets of a given set. To do so, you need to look at the number of elements in the given set. In general, if you have [latex]n[/latex] elements in your set, then there are [latex]2^{n}[/latex] subsets and [latex]2^{n}−1[/latex] proper subsets.

number of subsets and proper subsets set

If you have [latex]n[/latex] elements in your set:

  • Number of subsets: [latex]2^{n}[/latex]
  • Number of proper subsets: [latex]2^{n}−1[/latex]
Determine the total number of subsets and proper subsets of the set [latex]\{1,3,5,7\}[/latex].

Subsets of real numbers

The idea of subsets can also be applied to the sets of real numbers. For example, the set of all whole numbers is a subset of the set of all integers. The set of integers in turn is contained within the set of rational numbers.

We say the integers are a subset of the rational numbers. In fact that the integers are a proper subset of the rational numbers.