Set
A
group of numbers, variables, geometric
figures, or just about anything.
Sets are written using set braces {}. For example, {1,2,3}
is the set containing the elements 1, 2, and 3.
Note: Order does
not matter in a set. The sets {a,b,c} and {c,a,b} are the same
set. Repetition does not matter either, so {a,b} and {a,a,b,b,b}
are the same set.
See
also
Interval notation, set-builder
notation,
set intersection, union, set
subtraction, Venn diagrams, solution
set
Worked Example
Problem: Let A = {2, 4, 6, 8} and B = {1, 2, 3, 4}. Find the intersection A ∩ B and the union A ∪ B.
Step 1: Identify the elements that appear in both A and B. The numbers 2 and 4 are in both sets.
A∩B={2,4} Step 2: Combine all elements from A and B into one set. List each element only once, since sets ignore duplicates.
A∪B={1,2,3,4,6,8} Answer: The intersection is {2, 4} and the union is {1, 2, 3, 4, 6, 8}.
Another Example
Problem: Determine whether the following are the same set: P = {3, 5, 7, 5, 3} and Q = {7, 5, 3}.
Step 1: Remove duplicate elements from P. The distinct elements are 3, 5, and 7.
P={3,5,7} Step 2: Compare the distinct elements of P and Q. Both contain exactly the elements 3, 5, and 7, regardless of the order they are listed in.
Answer: Yes, P and Q are the same set because they contain exactly the same distinct elements.
Frequently Asked Questions
What is the difference between a set and a list?
A set contains only distinct elements and has no defined order, so {1, 2, 3} and {3, 2, 1} are the same set. A list (or sequence) can contain duplicates and treats order as meaningful, so (1, 2, 3) and (3, 2, 1) are different lists.
What is the empty set?
The empty set is the unique set that contains no elements at all. It is written as {} or with the symbol ∅. Every set contains the empty set as a subset. Set vs. Subset
A set is an entire collection of elements, such as A = {1, 2, 3, 4}. A subset is a set whose every element also belongs to another set. For example, {1, 3} is a subset of A because both 1 and 3 are in A. Every set is a subset of itself, and the empty set is a subset of every set.
Why It Matters
Sets are the foundation of nearly all modern mathematics. Concepts like functions, probability, and number systems are all defined using sets. In everyday math courses, you use set notation to describe solution sets for equations and inequalities, define domains and ranges of functions, and organize data in probability problems with Venn diagrams.
Common Mistakes
Mistake: Thinking that the order of elements in a set matters, for example treating {1, 2, 3} and {3, 1, 2} as different sets.
Correction: Order never matters in a set. Two sets are equal if and only if they contain exactly the same elements, regardless of how those elements are listed.
Mistake: Counting duplicate entries as separate elements, for example saying {2, 2, 5} has three elements.
Correction: Duplicates are ignored in a set. The set {2, 2, 5} has only two distinct elements — 2 and 5 — so it is identical to {2, 5}.
