# Set

A set is a collection of mathematical objects. Mathematical objects can range from points in space to shapes, numbers, symbols, variables, other sets, and more. Each object in a set is referred to as an element. Below are a few examples of different types of sets.

• A = {a, b, c, d, e} is a set made up of lower-case letters.
• B = {n|n ∈ N, 1 ≤ n ≤ 10} is a set made up of the first 10 natural numbers.
• C = {green, blue, red, yellow, white, black, purple} is a set made up of colors.
• M = {Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec} is a set made up of the 12 months in a year.
• P = the set of polygons shown in the figure below:

• 𝕌 = the universal set that contains subsets A and B, shown in the venn diagram below.

## Ways to define a set

As can be seen in the list above, there are a number of different ways to define a set.

### Definition through rules

A set can be defined using rules to determine the elements of the set. The rules must be strict such that the set is well-defined. For example, "the set of all even integers," is a well-defined set. There is nothing ambiguous about the set because the even integers are well-defined.

In contrast to the above, "the set of all young people who like ice cream," is not a well-defined set. "Young people" is an ambiguous term that doesn't specify age while the degrees to which people "like" ice cream cannot be quantified.

### Roster notation

Roster notation defines a set using curly brackets to contain a list of the elements that make up the set, separated by commas. The set of all even integers can be defined in roster notation as:

A = {..., -6, -4, -2, 0, 2, 4, 6, ...}

### Set-builder notation

Set-builder notation is similar to roster notation in its use of brackets, but rather than listing elements, conditions expressed using specific symbols (described in the table below) are applied to a larger set in order to specify a smaller set. For example, the set of all even integers is a subset of the set of all integers, and can be expressed in set-builder notation as

A = {2a | a∈ℤ}

where a is an integer, and "|" is read as "such that."

Set-builder notation can also be expressed in other ways. For example, the set of all integers greater than 12 could be expressed as:

B = {b∈ℤ | b>12}

## Symbols used in set theory

There are many different symbols that are used within set theory. The table below includes some of the most common symbols.

Symbol Definition/meaning Example
{ } Indicates a collection of elements {1, 3, 7, 9}
Empty set - set contains no elements/td> {}
... Indicates that the set continues the pattern in the corresponding direction (towards negative (left) or positive (right) infinity) {..., -9, -7, -3, -1, 1, 3, 7, 9, ...}
| "Such that" a∈ℤ | a > 3 - "a is an integer such that a is greater than 3"
"and" or "intersection" A = {1, 2, 3, 4}
B = {4, 5, 6}
A ∩ B = {4}
"or" or "union" A = {1, 2, 3, 4}
B = {4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6}
Subset - A is a subset of B if all its elements are included in B {1, 2} ⊆ {1, 2, 3, 4, 5}
{1, 2, 3, 4, 5} ⊆ {1, 2, 3, 4, 5}
Proper/strict subset - A is a proper subset of B if A is a subset of B, but not equal to B {1} ⊂ {1, 2, 3, 4, 5}
{1, 2} ⊂ {1, 2, 3, 4, 5}
{1, 2, 3} ⊂ {1, 2, 3, 4, 5}
{1, 2, 3, 4} ⊂ {1, 2, 3, 4, 5}
Element of - indicates that the object on the left of the symbol is an element of the object on the right x∈ℚ - "x is an element of the rational numbers"
𝕌 Universal set - the set of all possible values A = {1, 2}
B = {3, 4, 5}
𝕌 = {1, 2, 3, 4, 5}
Ac or A' Complement - all the elements not in set A 𝕌 = {1, 2, 3, 4, 5}
A = {1, 2, 3}
Ac = {4,5}
[a, b] Closed interval - values between a and b including a and b [1,4] = {1, 2, 3, 4} if only including integers
(a, b) Open interval - values between a and b not including a and b (1,4) = {2, 3} if only including integers
|A| Order/cardinality - number of elements in the set A = {3, 6, 7, 9}
|A| = 4
Natural numbers - only positive numbers with no decimals or fractions {1, 2, 3, ...}
Integers - all positive and negative numbers with no decimals or fractions, including 0 {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational numbers - a number that can be represented as a fraction comprised of integers
Real numbers - rational numbers and irrational numbers π, e, 3, ½, 0.25
Complex numbers - numbers made up of a real and imaginary component 4 + 2ⅈ