Learn the basics of Set Theory.

1. SET THEORY The Basics

2. SETS
What is a set?
A set is a collection of objects.
Why are sets important?
It is useful to understand the concept of sets because they play a
crucial role in the field of mathematics.
What does this lesson cover?
The notations, properties, and operations of sets.

3. EXAMPLE

4. DECK = SET OF CARDS
A deck is an example of a set
It is a set of cards
This is how we define the set:
D = {A♥, A♦, A♠, A♣, 2♥, 2♦, 2♠, 2♣, ··· J♥, J♦, J♠, J♣, Q♥, Q♦, Q♠, Q♣, K♥, K♦, K♠, K♣}

5. SET NOTATION
Rules:
1. Set names are capitalized – D is the name of the set in this case
2. The members of a set, known as elements, are separated by commas
3. Elements are enclosed in curly braces
4. The ellipsis (…) symbol is used to indicate the omission of elements – in
this example, the cards of all suits ranked 3-10
D = {A♥, A♦, A♠, A♣, 2♥, 2♦, 2♠, 2♣, ··· J♥, J♦, J♠, J♣, Q♥, Q♦, Q♠, Q♣, K♥, K♦, K♠, K♣}

6. ELEMENTS
A♥, A♦, A♠, A♣…
are examples of elements in the set D.
We use the  symbol to show membership in a set.
 is used if an item does not belong to a specific set.
Example:
A♥  D
14 ♥  D

7. SET EQUIVALENCE
2 sets are equal if they contain the same elements:
D = {2♥, 2♦, 2♠, 2♣, ··· J♥, J♦, J♠, J♣, Q♥, Q♦, Q♠, Q♣, K♥, K♦, K♠, K♣, A♥, A♦, A♠, A♣}
D = {A♥, A♦, A♠, A♣, 2♥, 2♦, 2♠, 2♣, ··· J♥, J♦, J♠, J♣, Q♥, Q♦, Q♠, Q♣, K♥, K♦, K♠, K♣}
Set order is irrelevant!

8. SUBSETS
A set A is considered a subset of set B if all the elements found in A
are in B.
Notation: A  B
If A is not equal to B it is considered a proper subset of B: A  B

9. SUBSET VS. PROPER SUBSET
D = standard deck
D  D
A = set of aces
A  D

10. CARDINALITY
Cardinality refers to the size of a set.
The set of standard playing cards, D, has a cardinality of 52 (excluding
jokers)
Notation: |D| = 52
The set of aces, A, has cardinality 4
Notation: |A| = 4

11. SET OPERATIONS
The most common set operations are:
I. UNION
II. INTERSECTION

12. UNION
Suppose we have another set:
C = Set of Face Cards (C  D)

13. UNION
The union of 2 sets is denoted: A  C
A  C = {A♥, A♦, A♠, A♣, J♥, J♦, J♠, J♣, Q♥, Q♦, Q♠, Q♣, K♥, K♦, K♠, K♣}
The elements in this resulting set are either in A or C.
The cards are either elements of the set of aces (A), or they are face cards
(C).

14. UNION: A  C
A C
Elements in A or C

15. INTERSECTION
Consider another set:
B = Set of cards of the club suit (B  D)

16. INTERSECTION
Performing the intersection operation of sets B and C:
B  C = {J♣, Q♣, K♣}
The elements contained in the intersection B  C , which we’ll rename as the set I, are
the elements found in sets B and C (have the clubs suit and are face cards).

17. INTERSECTION: B  C
B C
I = intersection of sets B and C
I

18. YOU NOW KNOW THE BASICS OF
SET THEORY
Congratulations on
completing this tutorial!