Sets

Set Theory is fundamental for understanding mathematical terms and logics. However, I won't cover all the topics in Set Theory. On this page, I'll introduce the basic definitions of sets, and on the next page, I'll cover operations on sets and Venn Diagrams which are powerful tools for visualizing these operations.

1. Definiton of a Set

A set is a well-defined collection of objects.

Here, 'well-defined' means that no matter who collects the objects, the resulting collections will always be the same.

Let us look at the following two examples:
1. the collection of good presidents in the U.S.
2. the collection of U.S. presidents who were over 65 years old at the start of their presidencies.

For the first collection, the definition of 'good' varies based on individual opinions. In other words, the resulting collections also will vary based on individuals. Thus, the first collection is not a set.

Now, let us consider the second collection. By googling, anyone can find the U.S. presidents over 65 years old at the start of their presidencies. The resulting collections will be the same for everyone. Thus, the second collection is a set.

Example 1. Determine whether each collection is a set.

(a) the collection of the distinct letters of 'happy'

(b) the collection of natural numbers between 3 and 7

(d) the collection of actors who won the Academy Awards for Best Supporting Actor from 2020 to 2024

Answer

An element is an object(or member) of a set.

Let us consider the set of distinct letters of 'happy.' Then the objects of the set are 'h,' 'a,' 'p,' and 'y.' In other words, 'h,' 'a,' 'p,' and 'y' are called elements of the set.

Example 2. List all the elements of each set.

(a) the set of the distinct letters of 'happy'

(b) the set of counting numbers between 3 and 7

Answer

2. Three Ways of Describing a Set

a. Roster Method

In the roster method, we list all the element of a set.

Let us write the set of distinct letters of 'happy' using the roster method. We can write it as

the set of h, a, p, y

or {h, a, p, y}

The braces { } are used to represent ‘the set of’ From now on, I will use the braces to represent 'the set of.'

Example 3. Write each set using the roster method.

(a) {counting numbers between 3 and 7}

(b) {actors who won the Academy Awards for Best Supporting Actor from 2020 to 2024}

Answer

b. Descriptive Method

In the descriptive method, we use a statement to represent all the elements of a set.

Let us write {2, 4, 6, 8} using the descriptive method. We can write it as

{even numbers between 1 and 9}, or

{even numbers from 2 to 8}, or

{even numbers between 0 and 10}.

As you can see, there is more than one way to describe a set using the descriptive method.

Example 4. Write each set using the descriptive method. The answers will vary based on individuals.

(a) {1, 2, 3, 4, 5, 6, 7, ......}

(b) {a, p, l, e}

Answer

c. Set-Builder Notation

In set-builder notation, a set is described using a variable (commonly x) and conditions x must satisfy. The general form of set-builder notation is

{x | condition x must satisfy}

To use set-builder notation, we should understand the following notations:

1. the letter 'x' represents different elements of a set

2. the notation '|' means 'such that'

3. the notation '∈' means 'belongs to' or 'is contained in'

For example, we can write p ∈ {a, p, l, e} because p is an element of the set {a, p, l, e}.

Let us write the set {1, 2, 3, 4, 5, 6, 7, ......} using the set-builder notation. We can write it as

{x | x is a natural number}.

In the descriptive method, the above set is the set of all natural numbers.

Alternatively, we can write it as

{x | x ∈ }, where is the set of all the natural numbers.

Example 5. Write each set using the set-builder notation. The answers will vary based on individuals.

(a) the set of multiples of 3.

(b) {-12, -8, -4, 0, 4, 8, 12, 15}

Answer

Example 6. Write each set using the roster method.

(a) {multiples of 10 between 10 and 190}

(b) {x | x = 8k, k = -3, -2, -1, 0, 1, 2, 3, 4}

Answer

3. Equal and Equivalent Sets

a. Equal Sets

The two sets A and B are said to be equal if they have exactly the same elements. This is written as A = B.

Let us consider the following two sets defined by

{3, 6, 9, 12} and {x | x = 3k, k = 1, 2, 3, 4}.

If we write the set {x | x = 3k, k = 1, 2, 3, 4} using the roster method, we get {3, 6, 9, 12}. Thus, the two sets {3, 6, 9, 12} and {x | x = 3k, k = 1, 2, 3, 4} are equal.

Example 7. Determine whether the following sets are all equal. If they are not all equal, specify which ones are equal.

(a) {a, b, c, d}

(b) {a, b, b, c, d}

(d) {d, a, b, d, a}

(e) {c, a, b, d}

Answer

From the above example, we observe that the order of elements in a set doesn't affect the set itself. For instance, {a, b, c, d} is equal to {c, a, b, d}. Additionally, if an element is repeated, then it is generally not listed more than once. For instance, we write the set {a, b, b, c, d} as {a, b, c, d}.

b. Equivalent Sets

The two sets A and B are said to be equivalent if they have the same number of elements. This is written as A ↔ B.

Let us consider the following two sets

{1, 2, 3} and {2, 3, 4}.

The two sets are not equal because the element 1 is not contained in the set {2, 3, 4}. (Similarly, we can see that the element 4 doesn't belong to the set {1, 2, 3}.)

However, they are equivalent because both sets {1, 2, 3} and {2, 3, 4} have three elements, that is, the number of elements is the same to both sets.

Example 8. Determine whether the following sets are all equivalent. If they are not all equivalent, specify which ones are equivalent.

(a) {h, a, p, y}

(b) {2, 1, 3, 0}

Answer

The cardinality of a set is the number of elements in the set.

The cardinality of a set E is denoted by |E|. For example,

|{a, b, c, d}| = 4.

4. Universal Sets and the Empty Set

a. Universal Sets

A universal set is the set of all the elements under consideration,

denoted by U.

Let us consider the following example.

Suppose that a universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be given. Next, we will find a set of even numbers. Since the universal set is given, we will only consider the numbers from 1 to 10. Thus, the set of even numbers is {2, 4, 6, 8, 10}, not {2, 4, 6, 8, 10, 12, 14, ........}. This shows the concept of a universal set.

b. Empty Set

The empty set is the set that contains no elements, denoted by { } or ∅.

Since the empty set has no elements, the cardinality is zero, that is, |∅| = 0.

5. Subsets

a. Subsets

A is said to be a subset of B if every element of A is an element of B.

It is written by A ⊆ B.

Let us consider the following three sets:

A = {a, b, e}, B = {b, c, d, e}, C = {b, c, e}.

The determine whether (1) A is a subset of B; (2) B is a subset of C; (3) C is a subset of B; (4) A is a subset of A itself.

(1) A is not a subset of B because the element 'a' is not contained in B.

(2) B is not a subset of C because the element 'd' is not contained in C.

(3) C is a subset of B because all the elements in C are contained in B.

(4) A is a subset of A itself because all the elements in A are contained in A.

A is said to be a proper subset of B if A ⊆ B and A B.

It is written by A ⊂ B.

In the above example, we've seen that C is a subset of B, but is not equal to B. Thus, C is a proper subset of B. Also, A is a subset of and equal to itself. Thus, A is not a proper subset of A.

Note: the empty set ∅ is a subset of any set.

Here is another definition about subsets.

A is an improper subset of A.

Now, I'll introduce a property about the number of subsets of a set.

Number of Subsets

The number of subsets of a set of cardinality 'n' is .