1. Set Theory

2. Section 1
Set theory is the branch of mathematics
that studies sets, which are collections of
objects. Although any type of objects can
be collected into a set, set theory is
applied most often to objects that are
relevant to mathematics.

3. Elements/Members
Individual objects contained in the
collection
Ex: {x,a,p, or d}

4. Set-Builder Notation
We represent sets by listing elements or
by using set-builder notation
Example:
C= {x : x is a carnivorous animal}

5. Well Defined
A set is well defined if we are able to tell
whether any particular object is an
element of that set.
A= {x : x is a winner of an Academy
Award}
T= {x : x is tall}

6. Empty or Null Set
The set that contains no elements is called
the empty set or null set. This is labeled by
a symbol that has a 0 with a / going
through it.

7. Universal Set
The universal set is the set of all elements
under consideration in a given discussion.
It is often described by using the capital
letter U.

8. Cardinal Numbers
The actual number of elements in a Set is
its cardinal number. It is described by
using n(A).

9. Finite and Infinite Numbers
Set can either be finite or infinite
depending on the whole number. If a sets
cardinal number is a whole number then it
is finite. If it is not, then it is infinite.

10. 1.3 The Language of Sets Problems
 Use Set notation to list all the elements of each sets.
 M= The months of the year
M= {January, February, March, April, May,
June…}
 P=Pizza Toppings
P={pepperoni, cheese, mushrooms, anchovies,…}
Anyone
ordered
pizza?

11. 1.3 The Language of Sets Problems
 Determine whether each set is Well Defined:
 {x:x lives in Michigan}
Well Defined
 {y:y has an interesting job}
Not Well Defined
 State Whether each set is finite or infinite.
 P={x:x is a planet in our solar system}
Finite
 N={1,2,3,…}
Infinite

12. Equal Sets
Two sets can be considered equal if they
have the exact same members in them. It
would be written as A=B.
If A and B were not equal then it would be
A = B.

13. Subset
A subset would occur if every element of
one set is also an element in another set.
Using A and B, we could say that all the
elements of A were also in B too, and it
would be wrote as A then a sideways U
underlined with B after.

14. Proper Subsets
Using A and B, Set A would be a proper
subset of B if A ¢ B but A = B.

15. 1.3 The Language of Sets Problems
 1) A={x : x lives in Raleigh} B={x : x lives in North Carolina}
 Is A a subset of B?
Answer: Yes, A is a subset of B because, Raleigh lies
within North Carolina
 2) A={1,2,3} B={1,2,3,5,6,7)
 Is A a subset of B?
Answer: Yes, A is a subset of B because the numbers in A
are in B
 3) A={1,2,3,4} B={1,2,3,5,6,7,8}
 Is A a subset of B?
Answer: No, because 4 is not incuded in set B.

16. Union
The union of two sets would be wrote as A
U B, which is the set of elements that are
members of A or B, or both too.
Using set-builder notation,
A U B = {x : x is a member of A or X is a
member of B}

17. Intersection
Intersection are written as A ∩ B, is the set
of elements that are in A and B.
Using set-builder notation, it would look
like:
A ∩ B = {x : x is a member of A and x is a
member of B}

18. Complements
With A being a subset of the universal (U),
the complement of A (A’) is the set of
elements of U that are not elements of A.

19. Other Definitions
Venn diagram – a method of visualizing
sets using various shapes
Disjoint – If A ∩ B = 0, then A and B are
disjoint.
Difference: B – A; all the elements in B but
not in A
Equivalent sets – two sets are equivalent if
n(A) = n(B).