1. Discrete Mathematics R. Johnsonbaugh
Chapter 2
The Language of Mathematics

2. 2.1 Sets Set = a collection of distinct unordered objects
Members of a set are called elements
How to determine a set
Listing:
Example: A = {1,3,5,7} = {7, 5, 3, 1, 3}
Description
Example: B = {x | x = 2k + 1, 0 < k < 30}

3. Finite and infinite sets
Examples:
A = {1, 2, 3, 4}
B = {x | x is an integer, 1 < x < 4}
Infinite sets
Z = {integers} = {…, -3, -2, -1, 0, 1, 2, 3,…}
S={x| x is a real number and 1 < x < 4} = [0, 4]

4. Some important sets The empty set  = { } has no elements.
Also called null set or void set.
Universal set: the set of all elements about which we make
assertions.
Examples:
U = {all natural numbers}
U = {all real numbers}
U = {x| x is a natural number and 1< x<10}

5. Cardinality
Cardinality of a set A (in symbols |A|) is the number of elements in A
Examples:
If A = {1, 2, 3} then |A| = 3
If B = {x | x is a natural number and 1< x< 9}
then |B| = 9
Infinite cardinality
Countable (e.g., natural numbers, integers)
Uncountable (e.g., real numbers)

6. Subsets
X is a subset of Y if every element of X is also contained in Y (in symbols X  Y)
Equality: X = Y if X  Y and Y  X,
i.e., X = Y whenever x  X, then x  Y,
and whenever x  X, then x  X
X is a proper subset of Y if X  Y but Y  X
Observation:  is a subset of every set

7. Power set
The power set of X is the set of all subsets of X, in symbols P(X),i.e. P(X)= {A | A  X}
Example: if X = {1, 2, 3},
then P(X) = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
Theorem 2.1.4: If |X| = n, then |P(X)| = 2n.
See proof by induction in textbook

8. Set operations: Union and Intersection
Given two sets X and Y
The union of X and Y is defined as the set
X  Y = { x | x  X or x  Y}
The intersection of X and Y is defined as the set
X  Y = { x | x  X and x  Y}
Two sets X and Y are disjoint if X  Y = 

9. Complement and Difference
The difference of two sets
X – Y = { x | x  X and x  Y}
The difference is also called the relative complement of Y in X
Symmetric difference
X Δ Y = (X – Y)  (Y – X)
The set of all elements that belong to X or to Y but not both X and Y.
The complement of a set A contained in a universal set U is the set Ac = U – A
In symbols Ac = U - A

10. Example If X={1, 4, 7, 10}, Y={1, 2, 3, 4, 5} X  Y = X  Y = X – Y =
X Δ Y = (how else can you write this?)

11. Example
If X={1, 4, 7, 10}, Y={1, 2, 3, 4, 5}
X  Y = {1, 2, 3, 4, 5, 7, 10}
X  Y = {1, 4}
X – Y = {7, 10}
Y – X = {2, 3, 5}
X Δ Y = (X – Y)  (Y – X) = {2, 3, 5, 7, 10}

12. Venn diagrams A Venn diagram provides a graphic view of sets
Set union, intersection, difference, symmetric difference and complements can be easily and visually identified U C A B

13. Properties of set operations
Theorem : Let U be a universal set, and
A, B and C subsets of U. The following properties
hold:
Associativity: (A  B)  C = A  (B  C)
(A  B)  C = A  (B C)
Commutativity: A  B = B  A
A  B = B  A

14. Properties of set operations
Distributive laws:
A(BC) = (AB)(AC)
A(BC) = (AB)(AC)
Identity laws:
AU=A A = A
Complement laws:
AAc = U AAc = 

15. Properties of set operations
Idempotent laws:
AA = A AA = A
Bound laws:
AU = U A = 
Absorption laws:
A(AB) = A A(AB) = A

16. Properties of set operations
Involution law:
(Ac)c = A
0/1 laws: c = U Uc = 
De Morgan’s laws for sets:
(AB)c = AcBc
(AB)c = AcBc

17. Addition Principle A.K.A The Inclusion-Exclusion Principle
If A and B are finite sets then,
| A  B | = |A| + |B| -
| A  B | U
A  B A B

18. Addition Principle for Disjoint Sets
| A  B  C | =
|A| + |B| + |C| - |A  B| - |B  C| - |A  C| + |A  B  C|
A = { a, b, c, d, e } B = { a, b, e, g, h } C = { b, d, e, g, h, k, m, n}
A company wants to hire 25 programmers to handle systems
Programming jobs and 40 programmers for applications programming.
Of those hired, ten will be expected to perform jobs of both types.
How many programmers must be hired?

19. One more example
A survey was taken on methods of commuter travel. Each respondent was asked to check BUS, TRAIN, or CAR as a major mode of traveling. More than one answer is allowed. The results are:
BUS TRAIN CAR 100
BUS and TRAIN BUS and CAR 15
TRAIN and CAR All three modes 5
How many people drank coffee while traveling?
How many people completed a survey form

20. 2.2 Functions
A function f from X to Y (in symbols f : X  Y) is a relation from X to Y such that Dom(f) = X and if two pairs (x,y) and (x,y’)  f, then y = y’
Example:
Dom(f) = X = {a, b, c, d},
Rng(f) = {1, 3, 5}
f(a) = f(b) = 3, f(c) = 5, f(d) = 1.

21. Domain and Range Domain of f = X Range of f =
{ y | y = f(x) for some x X}
A function f : X  Y assigns to each x in Dom(f) = X a unique element y in Rng(f)  Y.
Therefore, no two pairs in f have the same first coordinate.

22. Algebraically speaking
Note that such definitions on functions are consistent with what you have seen in your Calculus courses.
function
not a function
 1 intersection
 violations when > 1

23. Modulus operator
Let x be a nonnegative integer and y a positive integer
r = x mod y is the remainder when x is divided by y
Examples:
1 = 13 mod 3
6 = 234 mod 19
4 = 2002 mod 111
Basically, remove the complete y’s and count what’s left
mod is called the modulus operator

24. One-to-one functions A function f : X  Y is one-to-one 
for each y  Y there exists at most one x  X with f(x) = y.
(therefore, f(x) = c is out of play)
Alternative definition: f : X  Y is one-to-one  for each pair of distinct elements x1, x2  X there exist two distinct elements y1, y2  Y such that f(x1) = y1 and f(x2) = y2.
Examples:
1. The function f(x) = 2x from the set of real numbers to itself is one-to-one
2. The function f : R  R defined by f(x) = x2 is not one-to-one, since for every real number x, f(x) = f(-x).

25. Onto functions A function f : X  Y is onto (or, subjective) 
for each y  Y there exists at least one x  X with f(x) = y, i.e. Rng(f) = Y.
Example: The function f={1,a),(2,c),(3,b)} from X={1,2,3} to Y={a,b,c} is 1-to-1 and onto. If Y={a,b,c,d}, then still 1-to-1, but not onto.
Example: The function f(x) = ex from the set of real numbers to itself is not onto Y = the set of all real numbers. However, if Y is restricted to Rng(f) = R +, the set of positive real numbers, then f(x) is onto. Why?
Look at the several visual examples illustrated in the textbook

26. Bijective functions
A function f : X Y is bijective  f is one-to-one and onto
Examples:
1. Is A linear function f(x) = ax + b a bijective function from the set of real numbers to itself. Why?
2. Is the function f(x) = x3 a bijective from the set of real numbers to itself. Why?

27. Inverse function Given a function y = f(x), the inverse f -1 is the
set {(y, x) | y = f(x)}
The inverse f -1 of f is not necessarily a function
Example: if f(x) = x2, then f -1 (4) = 4 = ± 2, not a unique
value and therefore f is not a function
However, if f is a bijective function, it can be shown that f -1 is a function
See Example

28. Exponential and logarithmic functions
Let f(x) = 2x and g(x) = log 2 x = lg x
f ◦ g(x) = f(g(x)) = f(lg x) = 2 lg x = x
g ◦ f(x) = g(f(x)) = g(2x) = lg 2x = x
Exponential and logarithmic functions are inverse functions

29. Composition of functions
Given two functions g : X  Y and f : Y  Z,
the composition f ◦ g is defined as follows:
f ◦ g (x) = f(g(x)) for every x  X.
Example:
g(x) = x2 – f(x) = 3x + 5
g ◦ f(x) = g(f(x)) = g(3x + 5) = (3x + 5)2 - 1
Composition of functions is associative
f ◦ (g ◦h) = (f ◦ g) ◦ h
In general, it is not commutative
f ◦ g  g ◦ f.

30. Binary operators
A binary operator on a set X is a function f that associates a single element of X to every pair of elements in X, i.e. f : X x X  X and f(x1, x2)  X for every pair of elements x1, x2.
Examples of binary operators are addition,subtraction and multiplication of real numbers, taking unions or intersections
of sets, concatenation of two strings over a set X, etc.

31. Unary operators
A unary operator on a set X associates to each single element of X
one element of X.
Examples:
Let X = U be a universal set and P(U) the power set of U
Define f : P(U)  P(U) the function defined by f (A) = A‘
the set complement of A in U, for every A  U.
Then f defines a unary operator on P(U).
(The operator here is the “complement” itself).

32. 2.3 Sequences and strings
A sequence is an ordered list of numbers, usually defined according to a formula function, sn, n = 1, 2, 3,... is the index of the sequence
If s is a sequence {sn| n = 1, 2, 3,…},
s1 denotes the first element,
s2 the second element,…
sn the nth element…
{n} is called the indexing set of the sequence. Usually the indexing set is N (natural numbers) or an infinite subset of N.

33. Examples of sequences Let s = {sn} be the sequence defined by
sn = 1/n , for n = 1, 2, 3,…
The first few elements of the sequence are:
1, ½, 1/3, ¼, 1/5,1/6,…
sn = n2 + 1, for n = 1, 2, 3,…
The first few elements of s are:
2, 5, 10, 17, 26, 37, 50,…

34. Increasing and decreasing
A sequence s = {sn} is said to be
increasing if sn < sn+1
decreasing if sn > sn+1, for every n = 1, 2, 3,…
Examples:
Sn = 4 – 2n, n = 1, 2, 3,… is decreasing:
2, 0, -2, -4, -6,…
Sn = 2n -1, n = 1, 2, 3,… is increasing:
1, 3, 5, 7, 9, …

35. Subsequences
A subsequence of a sequence s = {sn} is a sequence t = {tn} that consists of certain elements of s retained in the original order they had in s
Example: let s = {sn = n | n = 1, 2, 3,…}
1, 2, 3, 4, 5, 6, 7, 8,…
Let t = {tn = 2n | n = 1, 2, 3,…}
2, 4, 6, 8, 10, 12, 14, 16,…
t is a subsequence of s

36. Sigma notation  ak = a1 + a2 + … + am
If {an} is a sequence, then the sum m
 ak = a1 + a2 + … + am
k = 1
This is called the “sigma notation”, where the
Greek letter  indicates a sum of terms from
the sequence

37. Pi notation  ak = a1a2…am If {an} is a sequence, then the product m
This is called the “pi notation”, where the Greek letter  indicates a product of terms of the sequence

38. Strings
Let X be a nonempty set. A string over X is a finite sequence of elements from X.
Example: if X = {a, b, c}
Then  = bbaccc is a string over X
Notation: bbaccc = b2ac3
The length of a string  is the number of elements of  and is denoted by ||. If  = b2ac3 then || = 6
The null string is the string with no elements and is denoted by the Greek letter  (lambda). It has length zero.

39. More on strings Let X* = {all strings over X including }
Let X+ = X* - {}, the set of all non-null strings
Concatenation of two strings  and  is the operation on strings consisting of writing  followed by  to produce a new string 
Example:  = bbaccc and  = caaba,
then  = bbaccccaaba = b2ac4a2ba
Clearly, || = | | + ||