1. Sets

6. Set builder notation
N = {0,1,2,3,…} the “natural numbers”
R = reals
Z = {… -3,-2,-1,0,1,2,3,…}
Z+ = {1,2,3,4,5,…}
Q = the set of rational numbers

7. Set builder notation
S contains all elements from U (universal set)
That make predicate P true
Brace notation with ellipses (wee dots)

8. Set Membership
x is in the set S (x is a member of S)
y is not in the set S (not a member)

9. Am I making this up?

13. Set operators

14. Set empty
The empty set

15. The universal set U

16. Venn Diagram U B A
U is the universal set
A is a subset of B

17. Venn Diagram U B A
U is the universal set
A united with B

18. Venn Diagram U B A
U is the universal set
A intersection B

19. Cardinality of a set
The number of elements in a set (the size of the set)

20. Power Set
The set of all possible sets

21. Power Set
The set of all possible sets

22. Power Set
The set of all possible sets
We could represent a set with a bit string
0th element

23. Is this true for a set S?

24. Cartesian Product
A set of ordered tuples

25. (improper) Subset
is empty {} a subset of anything?
Is anything a subset of {}?
We have an implication, what is its truth table?
Note: improper subset!!

26. Consequently A is strictly smaller than B
(proper) Subset
Consequently A is strictly smaller than B
|A| < |B|

27. Equal sets?
Two show that 2 sets A and B are equal we need to show that
And we know that

28. Try This
Using set builder notation describe the following sets
odd integers in the range 1 to 9
the integers 1,4,9,16,25
even numbers in the range -8 to 8

29. Answers
Using set builder notation describe the following sets
odd integers in the range 1 to 9
the integers 1,4,9,16,25
even numbers in the range -8 to 8

30. How might a computer represent a set?
Remember those bit operations?

31. Computer Representation (possible}
How do we compute the following?
membership of an element in a set
union of 2 sets
intersection of 2 sets
compliment of a set
set difference (tricky?)

32. Power set
Try this
Compute the power set of
{1,2}
{1,2,3}
{{1},{2}} {}

33. Power set
Compute the power set of
{1,2}
{1,2,3}
{{1},{2}} {}
Think again … how might we represent sets?

34. Cartesian Product
A set of ordered tuples
note AxB is not equal to BxA
Just reminding you (and me)

35. Try This
Let A={1,2,3} and B={x,y}, find
AxB
BxA
if |A|=n and |B|=m what is |AxB|

36. My answer
Let A={1,2,3} and B={x,y}, find
AxB
BxA
if |A|=n and |B|=m what is |AxB|

43. Power Set PS(A) The thinking behind the code 1 11 01 111 011 101 00111 01
111
011
101
001 10
110
010
100
000 00
Go left: take
Go right: don’t take

44. Power Set PS(A) The thinking behind the code ▪▪▪ ▫▫▫ don’t select
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