0. Chapter 2: Basic Structures: Sets, Functions, Sequences and Sums
Sets (Section 2.1)
Set Operations (Section 2.2)
Functions (Section 2.3)
Sequences and Summations (Section 2.4)
© by Kenneth H. Rosen, Discrete Mathematics & its Applications, Sifth Edition, Mc Graw-Hill, 2007

1. Sets (2.1)
A set is a collection or group of objects or elements or members. (Cantor 1895)
A set is said to contain its elements.
There must be an underlying universal set U, either specifically stated or understood.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

2. Sets (2.1) (cont.) Notation: list the elements between braces:
S = {a, b, c, d}={b, c, a, d, d}
(Note: listing an object more than once does not change the set. Ordering means nothing.)
specification by predicates:
S= {x| P(x)},
S contains all the elements from U which make the predicate P true.
brace notation with ellipses:
S = { , -3, -2, -1},
the negative integers.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

3. Common Universal Sets Notation: Sets (2.1) (cont.) R = reals
N = natural numbers = {0,1, 2, 3, }, the counting numbers
Z = all integers = {. . , -3, -2, -1, 0, 1, 2, 3, 4, . .}
Z+ is the set of positive integers
Notation:
x is a member of S or x is an element of S:
x  S.
x is not an element of S:
x  S.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

4. Sets (2.1) (cont.)
Subsets
Definition: The set A is a subset of the set B, denoted A  B, iff
x [x  A  x  B]
Definition: The void set, the null set, the empty set, denoted , is the set with no members.
Note: the assertion x   is always false. Hence
x [x    x  B] is always true(vacuously). Therefore,  is a subset of every set.
Note: A set B is always a subset of itself.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

5. Example: If A = {a, b} then
Sets (2.1) (cont.)
Definition: If A  B but A  B the we say A is a proper subset of B, denoted A  B (in some texts).
Definition: The set of all subset of a set A, denoted P(A), is called the power set of A.
Example: If A = {a, b} then
P(A) = {, {a}, {b}, {a,b}}
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

6. A is finite and so is P(A). Useful Fact: |A|=n implies |P(A)| = 2n
Sets (2.1) (cont.)
Definition: The number of (distinct) elements in A, denoted |A|, is called the cardinality of A.
If the cardinality is a natural number (in N), then the set is called finite, else infinite.
Example: A = {a, b},
|{a, b}| = 2,
|P({a, b})| = 4.
A is finite and so is P(A).
Useful Fact: |A|=n implies |P(A)| = 2n
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

7. Sets (2.1) (cont.)
N is infinite since |N| is not a natural number. It is called a transfinite cardinal number.
Note: Sets can be both members and subsets of other sets.
Example:
A = {,{}}.
A has two elements and hence four subsets:
, {}, {{}}. {,{}}
Note that  is both a member of A and a subset of A!
Russell's paradox: Let S be the set of all sets which are not members of themselves. Is S a member of itself?
Another paradox: Henry is a barber who shaves all people who do not shave themselves. Does Henry shave himself?
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

8. Definition: The Cartesian product of A with B, denoted
Sets (2.1) (cont.)
Definition: The Cartesian product of A with B, denoted
A x B, is the set of ordered pairs {<a, b> | a  A  b  B}
Notation:
Note: The Cartesian product of anything with  is . (why?)
Example:
A = {a,b}, B = {1, 2, 3}
AxB = {<a, 1>, <a, 2>, <a, 3>, <b, 1>, <b, 2>, <b, 3>}
What is BxA? AxBxA?
If |A| = m and |B| = n, what is |AxB|?
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

9. Set Operations (2.2) (cont.)
Propositional calculus and set theory are both instances of an algebraic system called a
Boolean Algebra.
The operators in set theory are defined in terms of the corresponding operator in propositional calculus
As always there must be a universe U. All sets are assumed to be subsets of U
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

10. Set Operations (2.2) (cont.)
Definition: Two sets A and B are equal, denoted A = B, iff
x [x  A  x  B].
Note: By a previous logical equivalence we have
A = B iff x [(x  A  x  B)  (x  B  x  A)] or
A = B iff A  B and B  A
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

11. Set Operations (2.2) (cont.)
Definitions:
The union of A and B, denoted A U B, is the set {x | x  A  x  B}
The intersection of A and B, denoted A  B, is the set
{x | x  A  x  B}
Note: If the intersection is void, A and B are said to be disjoint.
The complement of A, denoted , is the set {x | (x  A)}
Note: Alternative notation is Ac, and {x|x  A}.
The difference of A and B, or the complement of B relative to A, denoted A - B, is the set A 
Note: The (absolute) complement of A is U - A.
The symmetric difference of A and B, denoted A  B, is the set
(A - B) U (B - A)

12. Set Operations (2.2) (cont.)
Examples: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A= {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}. Then
AB = {1, 2, 3, 4, 5, 6, 7, 8}
A  B = {4, 5}
= {0, 6, 7, 8, 9, 10}
= {0, 1, 2, 3, 9, 10}
A - B = {1, 2, 3}
B - A = {6, 7, 8}
AB = {1, 2, 3, 6, 7, 8}
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

13. Set Operations (2.2) (cont.)
Venn Diagrams
A useful geometric visualization tool (for 3 or less sets)
The Universe U is the rectangular box
Each set is represented by a circle and its interior
All possible combinations of the sets must be represented
Shade the appropriate region to represent the given set operation. U U B A A B C
For 2 sets
For 3 sets
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

14. Set Operations (2.2) (cont.)
Set Identities
Set identities correspond to the logical equivalences.
Example:
The complement of the union is the intersection of the
complements:
= 
Proof: To show:
x [x   x   ]
To show two sets are equal we show for all x that x is a member of one set if and only if it is a member of the other.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

15. Set Operations (2.2) (cont.)
We now apply an important rule of inference (defined later) called
Universal Instantiation
In a proof we can eliminate the universal quantifier which binds a variable if we do not assume anything about the variable other than it is an arbitrary member of the Universe. We can then treat the resulting predicate as a proposition.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

16. Then we can treat the predicates as propositions:
We say
'Let x be arbitrary.'
Then we can treat the predicates as propositions:
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

17. Set Operations (2.2) (cont.)
Hence
x   x  
is a tautology.
Since
• x was arbitrary
• we have used only logically equivalent assertions and definitions
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

18. Set Operations (2.2) (cont.)
we can apply another rule of inference called
Universal Generalization
We can apply a universal quantifier to bind a variable if we have shown the predicate to be true for all values of the variable in the Universe.
and claim the assertion is true for all x, i.e.,
x [x   x   ]
Q. E. D. (Latin phrase “Quod Erat Demonstrandum”)
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

19. Set Operations (2.2) (cont.)
Note: As an alternative which might be easier in some cases, use the identity
A = B  [A  B and B  A]
Example:
Show A  (B - A) = 
The void set is a subset of every set. Hence,
A  (B - A)  
Therefore, it suffices to show
A  (B - A)   or x [xA  (B - A)  x  ]
So as before we say 'let x be arbitrary’.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

20. Set Operations (2.2) (cont.)
Example (cont.)
Show xA  (B - A)  x   is a tautology.
But the consequent is always false.
Therefore, the antecedent better always be false also.
Apply the definitions:
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

21. Set Operations (2.2) (cont.)
Example (cont.)
Hence, because P  P is always false, the implication is a tautology.
The result follows by Universal Generalization.
Q. E. D.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

22. Set Operations (2.2) (cont.)
Union and Intersection of Indexed Collections
Let A1,A2 ,..., An be an indexed collection of sets.
Union and intersection are associative (because 'and' and 'or' are) we have:
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

23. Set Operations (2.2) (cont.)
Examples Let
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

24. Functions (2.3)
Definition: Let A and B be sets. A function (mapping, map) f from A to B, denoted f :AB, is a subset of A*B such that
x [x  A  y [y  B  < x, y > f ]]
and
[< x, y1 > f  < x, y2 >  f ]  y1 = y2
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

25. Functions (2.3) (cont.)
Note: f associates with each x in A one and only one y in B.
A is called the domain and
B is called the codomain.
If f(x) = y
y is called the image of x under f
x is called a preimage of y
(note there may be more than one preimage of y but there
is only one image of x).
The range of f is the set of all images of points in A under
f. We denote it by f(A).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

26. Functions (2.3) (cont.) If S is a subset of A then
f(S) = {f(s) | s in S}.
Example:
• f(a) = Z
• the image of d is Z
• the domain of f is A = {a, b, c, d}
• the codomain is B = {X, Y, Z}
• f(A) = {Y, Z}
• the preimage of Y is b
• the preimages of Z are a, c and d
• f({c,d}) = {Z} A B a X b Y c Z d
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

27. Functions (2.3) (cont.) Injections, Surjections and Bijections
Let f be a function from A to B.
Definition: f is one-to-one (denoted 1-1) or injective if preimages are unique.
Note: this means that if a  b then f(a)  f(b).
Definition: f is onto or surjective if every y in B has a preimage.
Note: this means that for every y in B there must be an x in A such that f(x) = y.
Definition: f is bijective if it is surjective and injective (one-to-one and onto).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

28. Functions (2.3) (cont.) Examples:
The previous Example function is neither an injection nor a surjection. Hence it is not a bijection. A B a X b Y c Z d
Surjection but not an injection
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

29. Functions (2.3) (cont.) A B A B a a V V b b W W c c X X d d Y Y
Injection & a surjection, hence a bijection Z
Injection but not a surjection
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

30. or the same cardinality.
Functions (2.3) (cont.)
Note: Whenever there is a bijection from A to B, the two sets must have the same number of elements
or the same cardinality.
That will become our definition, especially for infinite sets.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

31. Functions (2.3) (cont.) Examples:
Let A = B = R, the reals. Determine which are injections, surjections, bijections:
• f(x) = x,
• f(x) = x2,
• f(x) = x3,
• f(x) = x + sin(x),
• f(x) = | x |
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

32. Sorry! It gets worse before it gets better.
Functions (2.3) (cont.)
Let E be the set of even integers {0, 2, 4, 6, }.
Then there is a bijection f from N to E , the even nonnegative integers, defined by
f(x) = 2x.
Hence, the set of even integers has the same cardinality as the set of natural numbers.
OH, NO! IT CAN’T BE....E IS ONLY HALF AS BIG!!!
Sorry! It gets worse before it gets better.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

33. Functions (2.3) (cont.) Inverse Functions
Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f-1, is the function from B to A defined as
f-1(y) = x iff f(x) = y
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

34. Functions (2.3) (cont.) Example: Let f be defined by the diagram: AV V b b W W c c X X d d Y Y
Note: No inverse exists unless f is a bijection
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

35. Functions (2.3) (cont.) Definition: Let S be a subset of B. Then
f-1(S) = {x | f(x)  S}
Note: f need not be a bijection for this definition to hold.
Example: Let f be the following function: A B a X
f-1({Z}) = {c, d}
f-1({X, Y}) = {a, b} b Y c Z d
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

36. Functions (2.3) (cont.) Composition
Definition: Let f: B C, g: A B. The composition of f with g, denoted fg, is the function from A to C defined by
f  g(x) = f(g(x))
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

37. A g B f C a V h Examples: b W i c X j d Y A fg C a h b i c j d
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

38. Functions (2.3) (cont.)
If f(x) = x2 and g(x) = 2x + 1, then f(g(x)) = (2x+1)2 and g(f(x)) = 2x2 + 1
Definition:
The floor function, denoted f ( x) = x or f(x) = floor(x), is the largest integer less than or equal to x.
The ceiling function, denoted f ( x) = x or f(x) = ceiling(x), is the smallest integer greater than or equal to x.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

39. Functions (2.3) (cont.) Examples: 3.5 = 3, 3.5 = 4.
Note: the floor function is equivalent to truncation for positive numbers.
Example:
Suppose f: B  C, g: A  B and f  g is injective.
What can we say about f and g?
• We know that if a  b then f(g(a))  f(g(b)) since the composition is injective.
• Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point.
• Hence, g(a)  g(b)
It follows that g must be an injection.
However, f need not be an injection (you show).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

40. Sequences and Summations (2.4)
Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2, } to a set S.
Note: the sets
{0, 1, 2, 3, , k} and {1, 2, 3, 4, , k}
are called initial segments of N.
Notation: if f is a function from {0, 1, 2, . . .} to S we usually denote f(i) by ai and we write
where k is the upper limit (usually ).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

41. Sequences and Summations (2.4) (cont.)
Examples:
Using zero-origin indexing, if f(i) = 1/(i + 1). then the
Sequence
f = {1, 1/'2,1/3,1/4, } = {a0, a1, a2, a3, . . }
Using one-origin indexing the sequence f becomes
{1/2, 1/3, . . .} = {a1, a2, a3, . . .}
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

42. Sequences and Summations (2.4) (cont.)
Summation Notation
Given a sequence we can add together a subset of the sequence by using the summation and function notation
or more generally
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

43. Sequences and Summations (2.4) (cont.)
Examples:
Similarity for the product notation:

44. Sequences and Summations (2.4) (cont.)
Definition: A geometric progression is a sequence of the form
a, ar, ar2, ar3, ar4,
Your book has a proof that
(you can figure out what it is if r = 1).
You should also be able to determine the sum
• if the index starts at k vs. 0
• if the index ends at something other than n (e.g., n-1, n+1, etc.).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

45. Sequences and Summations (2.4) (cont.)
Cardinality
Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted
| A | = | B |,
if there exists a bijection from A to B.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

46. Sequences and Summations (2.4)
Definition: If a set has the same cardinality as a subset of the natural numbers N, then the set is called countable.
If |A| = |N|, the set A is countably infinite.
The (transfinite) cardinal number of the set N is
aleph null = 0.
If a set is not countable we say it is uncountable.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

47. Sequences and Summations (2.4)
Examples:
The following sets are uncountable (we show later)
The real numbers in [0, 1]
P(N), the power set of N
Note: With infinite sets proper subsets can have the same cardinality. This cannot happen with finite sets.
Countability carries with it the implication that there is a listing of the elements of the set.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

48. Sequences and Summations (2.4)
Definition: | A |  | B | if there is an injection from A to B.
Note: as you would hope,
Theorem:
If | A |  | B | and | B |  | A | then | A | = | B |.
This implies
if there is an injection from A to B
if there is an injection from B to A
then
there must be a bijection from A to B
This is difficult to prove but is an example of demonstrating existence without construction.
It is often easier to build the injections and then conclude the bijection exists.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

49. Sequences and Summations (2.4)
Example:
Theorem: If A is a subset of B then | A |  | B |.
Proof: the function f(x) = x is an injection from A to B.
Example: {0, 2, 5}|  0
The injection f: {0, 2, 5}  N defined by f(x) = x is shown below: …
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

50. Sequences and Summations (2.4)
Some Countably Infinite Sets
The set of even integers E ( 0 is considered even) is countably infinite. Note that E is a proper subset of N,
Proof: Let f(x) = 2x. Then f is a bijection from N to E
Z+, the set of positive integers is countably infinite. … …
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

51. Sequences and Summations (2.4)
The set of positive rational numbers Q+ is countably infinite.
Proof: Z+ is a subset of Q+ so |Z+| = 0  |Q+|.
Now we have to show that |Q+|  0.
To do this we show that the positive rational numbers with repetitions, QR, is countably infinite.
Then, since Q+ is a subset of QR, it follows that
|Q+|  0 and hence |Q+| = 0.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

52. Sequences and Summations (2.4)
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

53. Sequences and Summations (2.4)
The position on the path (listing) indicates the image of the bijective function f from N to QR:
f(0) = 1/1, f(1) = 1/2, f(2) = 2/1, f(3) = 3/1, and so forth.
Every rational number appears on the list at least once, some many times (repetitions).
Hence, |N| = |QR| =  Q. E. D
The set of all rational numbers Q, positive and negative, is countably infinite.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

54. Sequences and Summations (2.4)
The set of (finite length) strings S over a finite alphabet A is countably infinite.
To show this we assume that
A is nonvoid
There is an “alphabetical” ordering of the symbols in A
Proof: List the strings in lexicographic order:
all the strings of zero length,
then all the strings of length 1 in alphabetical order,
then all the strings of length 2 in alphabetical order,
etc.
This implies a bijection from N to the list of strings and hence it is a countably infinite set.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

55. Sequences and Summations (2.4)
For example:
Let A = {a, b, c}.
Then the lexicographic ordering of A is
{ , a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab,
aac, aba, ....} = {f(0), f(1), f(2), f(3), f(4), }
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

56. Sequences and Summations (2.4)
The set of all C programs is countable.
Proof: Let S be the set of legitimate characters which can appear in a C program.
A C compiler will determine if an input program is a syntactically correct C program (the program doesn't have to do anything useful).
Use the lexicographic ordering of S and feed the strings into the compiler.
If the compiler says YES, this is a syntactically correct C program, we add the program to the list.
Else we move on to the next string.
In this way we construct a list or an implied bijection from N to the set of C programs.
Hence, the set of C programs is countable Q. E. D.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

57. Sequences and Summations (2.4)
Cantor Diagonalization
An important technique used to construct an object which is not a member of a countable set of objects with (possibly) infinite descriptions
Theorem: The set of real numbers between 0 and 1 is uncountable.
Proof: We assume that it is countable and derive a contradiction.
If it is countable we can list them (i.e., there is a bijection from a subset of N to the set).
We show that no matter what list you produce we can construct a real number between 0 and 1 which is not in the list.
Hence, there cannot exist a list and therefore the set is not countable
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

58. Sequences and Summations (2.4)
It's actually much bigger than countable. It is said to have the cardinality of the continuum, c.
Represent each real number in the list using its decimal expansion.
e.g., 1/3 =
1/2 = =
If there is more than one expansion for a number, it doesn't matter as long as our construction takes this into account.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

59. Sequences and Summations (2.4)
THE LIST....
r1 = .d11d12d13d14d15d
r2 = .d21d22d23d24d25d
r3 = .d31d32d33d34d35d
...
Now construct the number x = .x1x2x3x4x5x6x
xi = 3 if dii  3
xi = 4 if dii = 3
(Note: choosing 0 and 9 is not a good idea because of the non uniqueness of decimal expansions.)
Then x is not equal to any number in the list.
Hence, no such list can exist and hence the interval (0,1) is uncountable Q. E. D.

60. Sequences and Summations (2.4)
An extra goody:
Definition: a number x between 0 and 1 is computable if there is a C program which when given the input i, will produce the ith digit in the decimal expansion of x.
Example: The number 1/3 is computable.
The C program which always outputs the digit 3, regardless if the input, computes the number.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

61. Sequences and Summations (2.4)
Theorem: There is exists a number x between 0 and 1 which is not computable.
There does not exist a C program (or a program in any other language) which will compute it!
Why? Because there are more numbers between 0 and 1 than there are C programs to compute them.
(in fact there are c such numbers!)
Our second example of the nonexistence of programs to compute things!
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions