What are Sets?
A set is a well-defined collection of distinct objects, called elements. Sets are a foundational building block of mathematics, used to describe groups of numbers, shapes, or any objects that share a property.
A set is a collection of distinct elements with no particular order, written with curly braces like A = {1, 2, 3}. Sets can be combined using union, intersection, and difference operations.
- •Contains every element that is in A, in B, or in both
- •Combines both sets together
- •|A ∪ B| = |A| + |B| − |A ∩ B|
- •Example: {1,2,3} ∪ {3,4} = {1,2,3,4}
- •Contains only elements that are in BOTH A and B
- •Finds what the sets share
- •|A ∩ B| ≤ min(|A|, |B|)
- •Example: {1,2,3} ∩ {3,4} = {3}
Try it: interactive calculator
Step-by-step worked examples
Let A = {1, 2, 3, 4} and B = {3, 4, 5}. Find A ∪ B and A ∩ B.
A ∪ B combines all elements: {1, 2, 3, 4, 5}
A ∩ B keeps only shared elements: {3, 4}
|A ∪ B| = 4 + 3 − 2 = 5, which matches the listed unionIn a class of 50 students, 30 study Math and 25 study Science, with 10 studying both. How many study at least one of the two?
|Math| = 30, |Science| = 25, |Math ∩ Science| = 10 |Math ∪ Science| = 30 + 25 − 10 = 45 So 45 students study at least one subject
Let A = {2, 4, 6} and B = {1, 3, 5}. Are A and B disjoint? Find |A ∪ B|.
A ∩ B = { } (no shared elements) → disjoint sets
|A ∪ B| = |A| + |B| − |A ∩ B| = 3 + 3 − 0 = 6Flashcards
Quick quiz
Q1.If A = {1,2,3} and B = {2,3,4}, what is A ∩ B?
Q2.If |A| = 5, |B| = 7, and |A ∩ B| = 2, what is |A ∪ B|?
Q3.What symbol represents the empty set?
Q4.If every element of A is also in B, what is the relationship?
The full card deck, worked steps and AI-tutor support for “What are Sets?” are in Notek — study by hand before your exam.
Common mistakes
Listing repeated elements in a set, like {1, 1, 2}. — Correct: Sets only contain distinct elements: {1, 1, 2} is written as {1, 2}.
Confusing union (∪) with intersection (∩). — Correct: Union combines everything; intersection keeps only what's shared.
Thinking order matters in a set, e.g. {1,2} ≠ {2,1}. — Correct: Sets are unordered — {1,2} and {2,1} are the exact same set.
Forgetting to subtract the overlap when counting a union. — Correct: Use |A ∪ B| = |A| + |B| − |A ∩ B| so shared elements aren't counted twice.
FAQ
What is a set in math?
A set is a well-defined collection of distinct elements, like A = {1, 2, 3}.
What is the formula for the union of two sets?
|A ∪ B| = |A| + |B| − |A ∩ B|, which avoids double-counting shared elements.
What are examples of sets?
The set of even numbers, the set of vowels {a,e,i,o,u}, or the set of students in a class.
How to calculate the size of a union of two sets?
Add the sizes of both sets, then subtract the size of their intersection to remove double-counted elements.