1. CS104 : Discrete Structures
4/21/2017
CS104 : Discrete Structures
Chapter I
Introduction & Logic Theory
Prepared by Dr. Zakir H. Ahmed

2. Introduction to Discrete Structures
4/21/2017
Introduction to Discrete Structures
Prepared by Dr. Zakir H. Ahmed

3. What is Discrete Structure?
Discrete Objects
Separated from each other (Opposite of continuous)
e.g., integers, people, house,
Vs. Continuous objects: e.g., real number
Discrete Structures
The abstract mathematical structures used to represent discrete objects and relationships between the objects e.g. sets, relations, graphs

4. Why do we study Discrete Structures?
Information is stored and manipulated by computers in a discrete fashion …
As a student in computer science major, you need to know the basic language and conceptual foundation for all of the computer science, i.e., Discrete Structures!
Discrete structure concepts are also widely used throughout math, science, engineering, economics, biology, etc., …
Get training for rational thought!

5. Uses of Discrete Structures in Computer Science
Networking
Database
Image Processing
Programming Languages
Compilers & Interpreters
Software Engineering
Artificial Intelligence
Computer Architecture
Operating Systems
Security & Cryptography
Advanced Algorithms & Data Structures
Graphics & Animation ……

6. Tentative Syllabus & Text Book
4/21/2017
Tentative Syllabus & Text Book
Logic Theory
Fundamental Structures
Proof Techniques
Basics of Counting
Graph Theory
Discrete Probability
Text Books
Discrete Mathematics and Its Applications, 7th edition by Kenneth H. Rosen, McGraw Hill, 2007.
Prepared by Dr. Zakir H. Ahmed

7. 4/21/2017
Logic Theory
Prepared by Dr. Zakir H. Ahmed

8. 4/21/2017
Propositional Logic
Proposition: A proposition is a declarative statement ( a statement that declares a fact) that is either TRUE or FALSE, but not both.
The area of logic that deals with propositions is called propositional logic.
Propositions
Not Propositions
1. Riyadh is the capital of Saudi Arabia
1. How many students in this class?
2. Every cow has 4 legs.
2. Bring me coffee!
= 32
= 7
3. X + 2 = 3
4. Y + Z = X
Prepared by Dr. Zakir H. Ahmed

9. Propositional Logic - Applications
4/21/2017
Propositional Logic - Applications
We are using propositional logic as a foundation for formal proofs.
Propositional logic is also the key to writing good code…you can’t do any kind of conditional (if) statement without understanding the condition you’re testing.
All the logical connectives we will discuss are also found in hardware and are called “gates.”
Prepared by Dr. Zakir H. Ahmed

10. Propositional Logic - Negation
4/21/2017
Propositional Logic - Negation
We use letters to denote propositional variables
Truth value of true proposition is denoted by T
Truth value of false proposition is denoted by F
Negation: Suppose p is a proposition. The negation of p is written p and has meaning: “It is not the case that p.”
The proposition p is read “NOT p”
Example: p: “Today is Friday”
p: “Today is NOT Friday”
Truth table for negation: p p F T
Notice that p is a proposition!
Prepared by Dr. Zakir H. Ahmed

11. Propositional Logic - Conjunction
4/21/2017
Propositional Logic - Conjunction
Conjunction corresponds to English “and.”
Conjunction: Let p and q be two propositions. The conjunction of p and q, denoted by p  q, is the proposition “p and q”. The p  q is true when both p and q are true, otherwise false.
Example: p: “Today is Friday”
q: “It is raining today”
p  q: “Today is Friday and it is raining today”
Truth table for conjunction: p q
p  q F T
Prepared by Dr. Zakir H. Ahmed

12. Propositional Logic - Disjunction
4/21/2017
Propositional Logic - Disjunction
Disjunction corresponds to English “or.”
Disjunction: Let p and q be two propositions. The disjunction of p and q, denoted by p  q, is the proposition “p or q”. The p  q is false when both p and q are false, otherwise true.
Example: p: “Today is Friday”
q: “It is raining today”
p  q: “Today is Friday or it is raining today”
Truth table for disjunction: p q
p  q F T
Prepared by Dr. Zakir H. Ahmed

13. Propositional Logic – Exclusive Or
4/21/2017
Propositional Logic – Exclusive Or
Exclusive Or: Let p and q be two propositions. The exclusive or of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true, otherwise false.
Truth table for Exclusive Or: p q
p ⊕ q F T
Prepared by Dr. Zakir H. Ahmed

14. Propositional Logic – Conditional Statement
4/21/2017
Propositional Logic – Conditional Statement
It corresponds to English “if p then q” or “p implies q.”
Conditional: Let p and q be two propositions. The conditional statement (implication) p  q is the proposition “if p, then q”. The conditional statement p  q is false when p is true and q is false, otherwise true.
Example: If it is raining, then it is cloudy.
If I am elected, then I will lower taxes.
If you get 100% in the final, then you will get Grade A+.
Truth table for implication: p q
p  q F T
Prepared by Dr. Zakir H. Ahmed

15. Propositional Logic - Special Definitions
4/21/2017
Propositional Logic - Special Definitions
Converse: q  p is converse of p  q.
Ex.: p  q: “If it is noon, then I am hungry.” q  p: “If I am hungry, then it is noon.”
Contrapositive: q  p is contapositive of p  q.
Ex.: p  q: “If it is noon, then I am hungry.” q  p: “If I am not hungry, then it is not noon.”
Inverse: p  q is inverse of p  q.
Ex.: p  q: “If it is noon, then I am hungry.”
p  q: “If it is not noon, then I am not hungry.”
p  q has same truth values as q  p
Prepared by Dr. Zakir H. Ahmed

16. Propositional Logic - Questions
4/21/2017
Propositional Logic - Questions
Q1: Give some examples of p  q, then tell what is
q  p
q  p
p  q
Q2: State the converse, contrapositive, and inverse of each of them
If it rains today, then I will stay at home
I come to class whenever there is going to be a examination
A positive integer is prime only if it has no divisors other than 1 and itself.
Prepared by Dr. Zakir H. Ahmed

17. Propositional Logic – Biconditional Statement
4/21/2017
Propositional Logic – Biconditional Statement
It corresponds to English “p if and only if q”.
Biconditional: Let p and q be two propositions. The biconditional statement (bi-implication) p ↔ q is the proposition “p if and only if q”. The biconditional statement p ↔ q is true when p and q have the same truth values, otherwise false.
p ↔ q has same truth value as (p q)  (q  p).
Example: p: “You can take the flight”
q : “You buy a ticket”
p ↔ q : “You can take the
flight if and only if
you buy a ticket” p q
p ↔ q F T
Prepared by Dr. Zakir H. Ahmed

18. Propositional Logic – Precedence
4/21/2017
Propositional Logic – Precedence
Precedence of Logical Operators:
A compound proposition that is:
(1) always true is called a tautology
(2) always false is called a contradiction
(3) neither a tautology nor a contradiction is called
contingency or satisfiable.
Operator
Precedence Λ ν → ↔ 1 2 3 4 5 p p
p  p
p  p F T
Prepared by Dr. Zakir H. Ahmed

19. Propositional Logic – Compound Propositions
4/21/2017
Propositional Logic – Compound Propositions
Construct truth table for (p  q) → (p  q).
Q3: Construct truth table for
(p  q) ↔ (p  q).
(p ⊕ q) → (p ⊕ q).
(p  q) → (p  q)
p  q
p  q q q p F T
Prepared by Dr. Zakir H. Ahmed

20. Propositional Equivalences – Logical Equivalences
4/21/2017
Propositional Equivalences – Logical Equivalences
Equivalence: The compound propositions p and q are logically equivalent if p↔q is a tautology. In other words, p and q are logically equivalent if their truth tables are the same. We write p  q.
Example: (p  q)  p  q.
Truth tables for (p  q) and p  q :
p  q q p
(p  q)
p  q q p T F
Prepared by Dr. Zakir H. Ahmed

21. Propositional Equivalences – Logical Equivalences
4/21/2017
Propositional Equivalences – Logical Equivalences
Example: p  q  p  q.
Truth tables for p  q and p  q:
Q4: Show that p  (q  r)  (p  q)  (p  r)
p  q p
p  q q p T F
Prepared by Dr. Zakir H. Ahmed

22. Propositional Equivalence – Prove the following Logical Equivalences
4/21/2017
Propositional Equivalence – Prove the following Logical Equivalences
Equivalence
Name
p Λ T  p and p ν F  p
Identity laws
p ν T  T and p Λ F  F
Domination laws
p ν p  p and p Λ p  p
Idempotent laws
¬( ¬ p)  p
Double negation law
p ν q  q ν p and p Λ q  q Λ p
Commutative laws
(p ν q) ν r  p ν (q ν r) and (p Λ q) Λ r  p Λ (q Λ r)
Associative laws
p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p  r)
Distributive laws
¬(p ν q)  ¬p Λ ¬q and ¬(p Λ q)  ¬p ν ¬q
De Morgan’s laws
p  (p  q)  p and p  (p  q)  p
Absorption laws
p ν ¬p  T and p Λ ¬p  F
Negation laws
Prepared by Dr. Zakir H. Ahmed

23. Propositional Equivalence – Prove the following Logical Equivalences
4/21/2017
Propositional Equivalence – Prove the following Logical Equivalences
Prepared by Dr. Zakir H. Ahmed

24. Propositional Logic – Translating English Sentences
4/21/2017
Propositional Logic – Translating English Sentences
The sentence “The automated reply cannot be sent when the file system is full” can be translated as
q → p
where p: “The automated reply can be sent ”
q: “The file system is full”
The sentence “You cannot drive a car if you are under 4 feet tall unless you are older than 16 years old” can be translated as
(q r) → p
where p: “You can drive a car ”
q: “You are under 4 feet tall ”
r: “You are older than 16 years old”
Prepared by Dr. Zakir H. Ahmed

25. Propositional Logic – Translating English Sentences
4/21/2017
Propositional Logic – Translating English Sentences
The sentence “You can access the Internet from campus only if you are a computer science major or you are not a freshman” can be translated as
p → (q  r)
where p: “You can access the Internet from campus”
q: “You are a computer science major”
r: “You are a freshman”
Q5: Translate the following sentences into logical expressions:
“Access is granted whenever the user has paid the subscription fee and enters a valid password”
“If the user has not entered a valid password but has paid the subscription fee, then access is granted”
Prepared by Dr. Zakir H. Ahmed

26. Propositional Equivalences – Logic and Bit Operations
4/21/2017
Propositional Equivalences – Logic and Bit Operations
Bit: A bit is a symbol with two possible values, namely, 0 (zero) and 1 (one).
A bit can be used to represent a truth value as 1 for T and 0 for F
Bit string: A bit string is a sequence of bits. The length of the string is number of bits in the string.
Example: is a bit string of length eight
We define the bitwise OR, AND, and XOR of two strings of same length to be the strings that have as their bits the OR, AND, and XOR of the corresponding bits in the two strings, respectively.
We use the symbols , , and ⊕ to represent bitwise OR, AND, and XOR, respectively.
Prepared by Dr. Zakir H. Ahmed

27. Propositional Equivalences – Logic and Bit Operations
4/21/2017
Propositional Equivalences – Logic and Bit Operations
Truth table for bitwise OR, AND, and XOR:
Q6: Find bitwise OR, AND, and XOR of the bit strings
and
and
and
x ⊕ y
x  y
x  y y x 1
Prepared by Dr. Zakir H. Ahmed

28. Predicates and Quantifiers - Predicates
4/21/2017
Predicates and Quantifiers - Predicates
The statements “x > 3”, “x = y + 3”, and “computer x is functioning properly” are not propositions
This section discusses how proposition can be produced from such statements
The statement “x > 3” has two parts –
the variables x, the subject of the statement
is greater than 3, the predicate, the property that the subject of the statement can have
We denote P(x) = “x > 3”, where P denotes the predicate “is greater than 3” and x is the variable
Once we assign a value to x, it becomes proposition
Prepared by Dr. Zakir H. Ahmed

29. Predicates and Quantifiers - Predicates
4/21/2017
Predicates and Quantifiers - Predicates
Def: A predicate, or propositional function, is a function that takes some variable(s) as arguments and returns True or False.
Example: Let P(x) = “x > 3”, what are the truth values of P(2) and P(4)?
Sol: P(2) = “2 > 3” is false and P(4) = “4 > 3” is true
Example: Let Q(x, y) = “x = y + 3”, what are the truth values of Q(1, 2) and Q(3, 0)?
Sol: Q(1,2)= “1=2+3” is false, Q(3,0)= “3=0+3” is true
Def: A statement of the form P(x1, x2, …., xn) is the value of the propositional function P at the n-tuple (x1, x2, …., xn), and P is called n-ary predicate
Prepared by Dr. Zakir H. Ahmed

30. Predicates and Quantifiers- Predicates
4/21/2017
Predicates and Quantifiers- Predicates
Example: Consider the statement
if x > 0 then x := x + 1.
Here P(x) = “x > 0”. If P(x) is true for the value of x, then the assignment statement x := x + 1 is executed, x is increased by 1. If P(x) is false for the value of x, then the assignment statement is not executed, x remains same
The statements that describe valid input are known as preconditions, and the conditions that the output should satisfy when it has run are known as postconditions
Q7: Let R(x, y, z) = “x + y = z”, what are the truth values of R(1, 2, 3) and R(0, 0, 1)?
Prepared by Dr. Zakir H. Ahmed

31. Predicates and Quantifiers - Quantifiers
4/21/2017
Predicates and Quantifiers - Quantifiers
Another way of changing a predicate into a proposition is called quantification
In English, the words all, some, many, none, and few are used in quantifications
Two types of quantifications:
Universal quantifications – a predicate is true for every element under consideration
Existential quantifications – there is one or more element under consideration for which predicate is true
The area of logic that deals with predicates and quantifiers is called predicate calculus
Prepared by Dr. Zakir H. Ahmed

32. Predicates and Quantifiers – Universal Quantifiers
4/21/2017
Predicates and Quantifiers – Universal Quantifiers
Def: The universal quantification of P(x) is the statement “P(x) for all values of x in the domain”. The notation x P(x) denotes the universal quantification of P(x). Here  is called universal quantifier. An element for which P(x) is false is called a counterexample of x P(x).
Quantifiers:
When False?
When True?
Statement
There is an x for which P(x) is false
P(x) is false for every x
P(x) is true for every x
There is an x for which P(x) is true
x P(x)
 x P(x)
Prepared by Dr. Zakir H. Ahmed

33. Predicates and Quantifiers – Universal Quantifiers
4/21/2017
Predicates and Quantifiers – Universal Quantifiers
Example: Let P(x) = “x+1 > x”. What is the truth value of the quantification x P(x), where the domain consists of all real numbers?
Sol: Because P(x) is true for all real numbers x, the quantification x P(x) is true.
Example: Let Q(x) = “x < 2”. What is the truth value of the quantification x Q(x), where the domain consists of all real numbers?
Sol: Because Q(x) is not true for every real number x, because, for instance, Q(3) is false. That is, x=3 is a counterexample for the statement x Q(x).
Prepared by Dr. Zakir H. Ahmed

34. Predicates and Quantifiers – Universal Quantifiers
4/21/2017
Predicates and Quantifiers – Universal Quantifiers
Q8: Let P(x) = “x2 > 0”. What is the truth value of the quantification x P(x), where the domain consists of all real numbers?
Q9: Let Q(x) = “x2 < 10”. What is the truth value of the quantification x Q(x), where the domain consists of positive integers less than 4?
Q10: Let R(x) = “x2 ≥ x”. What is the truth value of the quantification x R(x), where the domain consists of all real numbers? What is the truth value of the quantification if the domain consists of all integers?
Prepared by Dr. Zakir H. Ahmed

35. Predicates and Quantifiers – Existential Quantifiers
4/21/2017
Predicates and Quantifiers – Existential Quantifiers
Def: The existential quantification of P(x) is the statement “There exists an element x in the domain such that P(x)”. The notation  x P(x) denotes the existential quantification of P(x). Here  is called existential quantifier.
Example: Let P(x) = “x > 3”. What is the truth value of the quantification x P(x), where the domain consists of all real numbers?
Sol: Because P(x) is sometimes true - for instance, when x=4 – the existential quantification of P(x), which is x P(x), is true.
Prepared by Dr. Zakir H. Ahmed

36. Predicates and Quantifiers – Existential Quantifiers
4/21/2017
Predicates and Quantifiers – Existential Quantifiers
Example: Let P(x) = “x = x+1”. What is the truth value of the quantification  x P(x), where the domain consists of all real numbers?
Sol: Because P(x) is false for every real number x, the quantification  x P(x) is false.
Q11: Let Q(x) = “x2 > 10”. What is the truth value of the quantification  x Q(x), where the domain consists of positive integers not exceeding 4?
Q12: Let R(x) = “x2 ≥ x”. What is the truth value of the quantification  x R(x), where the domain consists of all real numbers? What is the truth value of the quantification if the domain consists of all integers?
Prepared by Dr. Zakir H. Ahmed

37. 4/21/2017
Predicates and Quantifiers – Translating English into Logical Expression
Example: Express the statement “Every student in this class has studied calculus” using predicates and quantifiers.
Sol: We rewrite as “For every student in this class, the student has studied calculus”.
=> “For every student x in this class, x has studied calculus”.
Let P(x) = “x has studied calculus”, then our statement is represented by xP(x).
Q13: Express the statements “Some students in this class visited Dubai” and “Every student in this class has visited either Makkah or Madina” using predicates and quantifiers.
Prepared by Dr. Zakir H. Ahmed

38. Predicates and Quantifiers – Quantifiers with Restricted Domain
4/21/2017
Predicates and Quantifiers – Quantifiers with Restricted Domain
Example: What do the statements x < 0(x2 > 0), y≠0(y3 ≠ 0), and z > 0(z2 = 2) mean, where the domain in each case consists of all real numbers?
Sol: The statement x < 0(x2 > 0) states that for every number x with x < 0, x2 > 0. That is, it states “The square of a negative real number is positive”. The statement is the same as x (x < 0 → x2 > 0).
The statement y ≠ 0(y3 ≠ 0) states that for every real number y with y ≠ 0, we have y3 ≠ 0. That is, it states “The cube of every nonzero real is nonzero”. The statement is the same as y (y ≠ 0 → y3 ≠ 0).
Finally, the statement z > 0(z2 = 2) states that there exist a real number z with z > 0 such that z2=2. That is, it states “There is positive square root of 2”. The statement is the same as z (z > 0 ⌃z2 = 2).
Prepared by Dr. Zakir H. Ahmed

39. Predicates and Quantifiers – Quantifiers with Restricted Domain
4/21/2017
Predicates and Quantifiers – Quantifiers with Restricted Domain
Note: The restriction of a universal quantification is the same as the universal quantification of a conditional statement. For instance, x < 0(x2 > 0) is another way of expressing x (x < 0 → x2 > 0).
On the other hand, the restriction of an existential quantification is the same as the existential quantification of a conjunction. For instance, z>0(z2=2) can be expressed as z (z > 0 ⌃ z2 = 2).
Precedence of Quantifiers: The quantifiers  and  have higher precedence then all logical operators from propositional calculus. For example, xP(x) v Q(x) is the disjunction of xP(x) and Q(x). In other words, it means (xP(x)) v Q(x) rather than x (P(x) v Q(x))
Prepared by Dr. Zakir H. Ahmed

40. Predicates and Quantifiers – Negating Quantified Expression
4/21/2017
Predicates and Quantifiers – Negating Quantified Expression
Example: x P(x) means “P(x) is true for every x”. What about x P(x) ?
Sol: x P(x) = Not [“P(x) is true for every x.”]
=> “There is an x for which P(x) is not true.”
=> x P(x)
So, x P(x) is logically equivalent to x P(x).
Example: x P(x) means “P(x) is true for some x”. What about x P(x) ?
Sol: x P(x) = Not [“P(x) is true for some x.”]
=> “P(x) is not true for all x.”
=> x P(x)
So, x P(x) is logically equivalent to x P(x).
Prepared by Dr. Zakir H. Ahmed

41. Predicates and Quantifiers – Negating Quantified Expression
4/21/2017
Predicates and Quantifiers – Negating Quantified Expression
De Morgan’s laws for quantifiers:
When False?
When is Negation True?
Equivalent
Statement
Negation
There is an x for which P(x) is true
P(x) is true for every x
For every x,
P(x) is false
There is an x for which P(x) is false
x P(x)
 x P(x)
 x P(x)
x P(x)
Prepared by Dr. Zakir H. Ahmed

42. Predicates and Quantifiers – Negating Quantified Expression
4/21/2017
Predicates and Quantifiers – Negating Quantified Expression
Example: What are the negations of the statements “There is a religious man” and “All Saudi eat Rice”?
Sol: Let P(x) = “x is a religious”,
=> “There is a religious man”= xP(x), where the domain consists of all men
Now, xP(x) = xP(x) =“Every man is nonreligious”.
Next, let Q(x) = “x eats Rice”
=> “All Saudi eat Rice” = xQ(x), where the domain consists of all Saudi
Now,  xQ(x) = xQ(x) = “Some Saudi does not eat rice”.
Q14: What are the negations of the statements x (x2 > x) and x (x2 = 2)”?
Prepared by Dr. Zakir H. Ahmed

43. Predicates and Quantifiers – Negating Quantified Expression
4/21/2017
Predicates and Quantifiers – Negating Quantified Expression
Example: Let L(x) = “x is a lion”, F(x) = “x is fierce”, and C(x) = “x drinks coffee”. Show the quantifiers for the statements (i) All lions are fierce, (ii) Some lions don’t drink coffee, (iii) Some fierce creatures don’t drink coffee.
Sol: (i) x (L(x)  F(x)), ii) x (L(x)  C(x)), and
(iii) x (F(x)  C(x))
Q15: Let B(x) = “x is a hummingbird”, L(x) = “x is a large bird”, H(x) = “x lives on honey”, and R(x) = “x is richly colored”. Show the quantifiers for the statements (i) All hummingbirds are richly colored, (ii) No large birds live on honey, (iii) Birds that do not live on honey are dully colored, and (iv) Not all large birds live on honey.
Prepared by Dr. Zakir H. Ahmed

44. Predicates and Quantifiers – Logical Equivalences
4/21/2017
Predicates and Quantifiers – Logical Equivalences
Example: Show that x (P(x)  Q(x)) and xP(x)  x Q(x) are logically equivalent.
Sol: Let x (P(x)  Q(x)) is true
=> if a is in the domain, then P(a)  Q(a) is true
=> P(a) and Q(a) are true, for all a in the domain
=> xP(x) and xQ(x) are true
=> xP(x)  xQ(x) is true
So, they are logically equivalent.
Q16: Show that x(P(x)Q(x)) and x(P(x)  Q(x)) are logically equivalent.
Prepared by Dr. Zakir H. Ahmed

45. Nested Quantifiers – Definition
4/21/2017
Nested Quantifiers – Definition
Def: Two quantifiers are said to be nested if one is within the scope of the other, such as
xy (x + y = 0)
Note that xy (x + y = 0) is same as xQ(x), where Q(x) is yP(x, y), where P(x, y) is x + y = 0.
Example: Assume that the domain for the variables x and y consists of all real numbers.
The statement
xy (x + y = y + x)
says that x + y = y + x for all real numbers x and y. This is commutative law for addition of real numbers.
Prepared by Dr. Zakir H. Ahmed

46. Nested Quantifiers – Example
4/21/2017
Nested Quantifiers – Example
Example:
Similarly, the statement
xy (x + y = 0)
says that for every real number x there is a real number y such that x + y = 0. This states that every real number has an additive inverse.
xyz (x + (y + z) = (x + y) + z)
is associative law for addition of real numbers.
Prepared by Dr. Zakir H. Ahmed

47. Nested Quantifiers – Order of Quantifiers
4/21/2017
Nested Quantifiers – Order of Quantifiers
Example: Let P(x, y) be the statement “x + y = y + x”. What are the truth values of the quantifications xyP(x, y) and yxP(x, y), where the domain for all variables consists of all real numbers?
Sol: xyP(x, y) = “For all real numbers x, for all real numbers y, x + y = y + x”.
Since P(x, y) is true for all real numbers x and y, the proposition xyP(x, y) is also true.
Also, yxP(x, y) = “For all real numbers y, for all real numbers x, x + y = y + x” = xyP(x, y).
Note: Order of nested universal quantifiers in a statement without other quantifiers can be changed without changing meaning of quantified statement.
Prepared by Dr. Zakir H. Ahmed

48. Nested Quantifiers – Order of Quantifiers
4/21/2017
Nested Quantifiers – Order of Quantifiers
Example: Let Q(x, y) be the statement “x + y = 0”. What are the truth values of the quantifications yxQ(x, y) and xyQ(x, y), where the domain for all variables consists of all real numbers?
Sol: yxQ(x, y) = “There is a real number y such that for every real number x, x + y = 0”.
Since there is no real number y such that x + y = 0, for all real numbers x, the proposition yxQ(x, y) is false.
xyQ(x, y) = “For every real number x, there is a real number y such that x + y = 0”.
Given a real number x, there is a real number y, namely y = -x, such that x + y =0. So, xyQ(x, y) is true.
Prepared by Dr. Zakir H. Ahmed

49. Nested Quantifiers – Quantification of Two Variables
4/21/2017
Nested Quantifiers – Quantification of Two Variables
When False?
When True?
Statement
There is a pair x, y for which P(x, y) is false
P(x, y) is true for every pair x, y.
xyP(x, y)
yxP(x, y)
There is an x such that P(x, y) is false for every y
For every x there is a y for which P(x, y) is true
xyP(x, y),
For every x there is a y for which P(x, y) is false
There is an x such that P(x, y) is true for every y
xyP(x, y)
P(x, y) is false for every pair x, y.
There is a pair x, y for which P(x, y) is true
xyP(x, y)
yxP(x, y)
Q17: Let Q(x, y) = “x + y = y”. What are the truth values of the quantifications yxQ(x, y) and xyQ(x, y), where the domain for all variables consists of all real numbers?
Prepared by Dr. Zakir H. Ahmed

50. Nested Quantifiers – Order of Quantifiers
4/21/2017
Nested Quantifiers – Order of Quantifiers
Example: Let Q(x, y, z) be the statement “x + y = z”. What are the truth values of the quantifications xyzQ(x, y, z) and zxyQ(x, y, z), where the domain for all variables consists of all real numbers?
Sol: xyzQ(x, y, z) = “For all real numbers x and for all real numbers y there is a real number z such that x + y = z” is true.
The order of quantification is important here. Since zxyQ(x, y, z) = “There is a real number z such that for all real numbers x and for all real numbers y, x + y = z”, is false, because there is no z that satisfies the equation x + y = z for all for all real numbers x and y
Prepared by Dr. Zakir H. Ahmed

51. Nested Quantifiers – Translating Statements
4/21/2017
Nested Quantifiers – Translating Statements
Example: Translate the statement “The sum of two positive integers is always positive” into a logical expression.
Sol: Let x and y be two positive integers, then we can write the statement as “For all positive integers x and y, x + y is positive”. This can be expressed as xy((x > 0) ⌃ (y > 0) → (x + y > 0)), where the domain for both variables consists of integers. We can also express this as xy(x + y > 0), where the domain for both variables consists of positive integers.
Q18: Translate the statement “Every real number except zero has a multiplicative inverse” into a logical expression.
Prepared by Dr. Zakir H. Ahmed

52. Nested Quantifiers – Translating From Nested Quantifiers
4/21/2017
Nested Quantifiers – Translating From Nested Quantifiers
Example: Translate the statement
x(C(x) v y(C(y) ⌃ F(x, y)))
into English, where C(x)=“x has a computer”, F(x, y) = “x and y are friends”, and the domain for both x and y consists of all students in your class.
Sol: The statement says that for every student x in your class, x has a computer or there is a student y such that y has computer and x and y are friends. In other words, every student in your class has a computer or has a friend who has a computer.
Q19: Translate the statement
xyz((F(x, y) ⌃ F(x, z) ⌃ (y ≠ z)) → F(y, z))
into English, where F(a, b) means a and b are friends and the domain for x, y and z consists of all students in your class.
Prepared by Dr. Zakir H. Ahmed

53. 4/21/2017
End of
Chapter I
Prepared by Dr. Zakir H. Ahmed