A set is a well-defined collection of distinct objects, called elements or members. Sets are written with curly braces and named with capital letters, as in A = {2, 4, 6}. Order does not matter and repeated items count once, so {1, 2, 3} and {3, 1, 2} are the same set.
Each object is either in the set or out of it. The same set can be written as a list (roster form) or as a rule, and it has n(A) = 4 elements.
Formal definition and notation
A set is a collection of objects in which, for any object x, it is clear whether x belongs to the set (x ∈ A) or not (x ∉ A). Two sets are equal when they have exactly the same elements. A set can be described by listing its elements (roster form) or by a rule (set-builder form), and the number of elements in a finite set A is its cardinality, n(A).
Symbol
Read it as
Meaning
{}
the set of
curly braces hold the elements of a set
x ∈ A
x is an element of A
x belongs to the set A
x ∉ A
x is not an element of A
x does not belong to A
∅
the empty set
the set with no elements, also written {}
n(A)
n of A
the number of elements in A (its cardinality)
{x ∣ x > 5}
the set of all x such that x is greater than 5
set-builder notation: a rule instead of a list
Key facts
Well-defined means membership is never a matter of opinion: the even numbers form a set, but the good movies do not.
Order and repeats do not matter: the letters of BOOK form the set {B, O, K}, with 3 elements.
The empty set∅ has no elements. It is not the same as {0}, which has one element, the number 0.
A set is finite if you can count all its elements, like {1, 2, 3}, and infinite if you cannot, like the set of whole numbers.
An interval such as [2, 6] is a set of allreal numbers between 2 and 6, infinitely many. The set {2, 6} has just two elements.
Quick check: 3 questions
Question 1 of 3
How many elements are in the set of letters of the word LETTER?
Show answer and explanation
Answer: B. The distinct letters are L, E, T, R, so the set has 4 elements. 6 counts every letter including repeats; 3 and 5 miss or double-count a letter.
Question 2 of 3
Which set is equal to {a, b, c}?
Show answer and explanation
Answer: D. Equal sets have exactly the same elements in any order, so {c, a, b} is equal. A is missing c, B has an extra d, and C is a set whose single element is a set.
Question 3 of 3
Let C = {x ∣ x is an odd number, 1 < x < 10}. Which statement is true?
Show answer and explanation
Answer: A.C = {3, 5, 7, 9}, so 9 ∈ C. 1 and 10 are excluded by the strict inequalities, and n(C) = 4, not 5.
Set: frequently asked questions
What is an example of a set in math?
The set of days in a week, the set of even numbers and {2, 3, 5, 7}, the primes below 10, are all sets. Each has a clear rule for membership. A collection such as tall students is not a set in math, because people could disagree about who belongs.
What is the empty set?
The empty set, written ∅ or {}, is the set with no elements. For example, the set of whole numbers between 3 and 4 is empty. It has cardinality 0, and there is only one empty set. Do not write it as {∅}: that is a set with one element.
What is the difference between roster and set-builder notation?
Roster notation lists every element inside braces: {5, 10, 15, 20}. Set-builder notation states a rule: {x ∣ x is a multiple of 5, x ≤ 20}. Both describe the same set. Set-builder form is handy for large or infinite sets, where a full list is impossible.
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