1. More about Sets

2. What is a set? A set is a well defined collection of objects.
Is S = {some numbers} a set?
Is W = {Fred, George, Ron} a set?

3. What is in a set? The objects in a set are called elements or members.
We write 3S or
6 {1, 2, 3, …}
Also, Harry  W

4. How do you write a set?
They are enclosed by brackets and the elements are separated by commas. The series of dots (… ) are called ellipses and mean “the pattern continues”.

5. How do you write a set?
Sets can be written by rule S = {x| 1 < x < 10 and x  I}
or by roster S = {2, 3, 4, 5, …,9}

6. More details... Two sets are equal iff they contain the same elements.
If A = {2, 3, 4, … 11} and if B = {x|1 < x < 12 and x  I}, is A = B?

7. More details... U usually stands for the universal set.
We are usually given the universal set. It is often a set of numbers, such as I, E or R.

8. More details...
The complement of a set (written A’ or Ā ) contains everything in the universal set that isn’t in the set itself.
If U = {a, b, c, … f} and if A = {f, a, c, e}, find A’.

9. More details...
Set A is a subset of B ( A  B) if and only if every member of A is also in B.
If B = {a, b, c, … f} and if A = {f, a, c, e}, then A  B.
Is A’  B ?

10. More details...
Set A is a proper subset of B ( A  B) if and only if A  B but A  B.
A = {a, c, e}, B = {f, a, c, e} and C = {a, c, e, f},
Then A  B, and A  B.
Furthermore, B  C, but B  C.

11. More details...
A set with no elements is called the empty set and is written as {} or  .
The empty set is a subset of all sets.

12. More details...
The cardinality of a set is the number of elements in a set, and is written |S|.
If K = {3, 5, 8}, then
|K| = 3
If P is the set of all subsets of K, find |P|.

13. More details...
The cardinality of a set can only be found if the set is finite. But infinite sets contain an infinite number of elements.

14. More details...
The union of A and B (written AB) is the set that contains all elements of A and all elements of B.

15. More details...
The intersection of A and B (written AB) is the set that contains all elements that A and B have in common.

16. More details...
If the intersection of A and B is the empty set (written AB =  ), then A and B are disjoint sets.

17. More details...
The difference set of A and B (written A - B) is the set that contains all elements in A that are not in B.

18. More details...
The set of elements that are either in A or B, but not both is called the symmetric difference of A and B and is written A  B.

19. Example
If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5}
Is N  M ?
Is L  M?
Is N  M?

20. Example
If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5}
Write N’.
Write N O.
Write N  O.

21. Example
If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5}
Write |M  O|.
Write |O’|.
Write M - O.

22. Example
A B C D
AC  BD =
AC  BD =
AB  CD =

23. Example
A B C D F
CA  CF =
AD  CF =
AB  CF =

24. Any Questions?