1. Foundations of Discrete Mathematics
Chapter 1
By Dr. Dalia M. Gil, Ph.D.

2. Truth Tables
Statement means a statement of fact that is either true or false.
Let p and q be statements.

3. Truth Tables The compound statements “p or q” is written (p q)
“p and q” is written (p  q)
2. The truth values of these compound statements depend on those of p and q can be neatly summarized by tables called truth tables.

4. Truth Tables for (p  q) and (p  q)
 The values for p and q- each eather true (T) or false (F)
Possible values for p and q

5. A Truth Table for the Implication p  q
p  q is false if p is T (true) and q is F (false)

6. A Truth Table for the negation of p “¬p”
if p is T (true) then ¬p is F (false)

7. A Truth Table for Compound Statements
 The first two columns and the last column are the most important

8. A Truth Table for Double Implication
 p  q is true if both p and q are true or are false

9. A Truth Table for p  ¬(p  q)
 The procedure is to construct appropriate columns one by one until the answer is reached.

10. A Truth Table for p  ¬(p  q)
 First, obtain p  q

11. A Truth Table for p  ¬(p  q)
 Second, obtain ¬(p  q)

12. A Truth Table for p  ¬(p  q)
 Third, obtain p  ¬(p  q) from the first and the fourth columns

13. p  q  [((¬p)  r) (q  r)]
 There are three statements p, q, and r

14. p  q  [((¬p)  r) (q  r)]
 Obtain ¬p from p

15. p  q  [((¬p)  r) (q  r)]
Obtain the compound statement (¬p)  r

16. p  q  [((¬p)  r) (q  r)]
Obtain q  r

17. p  q  [((¬p)  r) (q  r)]
 Obtain the compound statement ((¬p)  r) (q  r)

18. p  q  [((¬p)  r) (q  r)]
 Obtain the statement p  q

19. p  q  [((¬p)  r) (q  r)]
 Obtain the compound statement p  q  [((¬p)  r) (q  r)]

20. p  q  [((¬p)  r) (q  r)]
p  q is false when
the values of p, q are T, F.

21.  [p ((q (¬r))  s]  [(¬t) (s  r)]
Where p, q, r, and s are all true while t is false.
Obtain the statements (¬r), (q (¬r)), and (p ((q (¬r))  s)

22.  [p ((q (¬r))  s]  [(¬t) (s  r)]
Obtain the statements (¬t), (s r), and (¬t)  (s  r)

23.  [p ((q (¬r))  s]  [(¬t) (s  r)]
Obtain the compound statement [p ((q (¬r))  s]  [(¬t) (s  r)]

24.  [p ((q (¬r))  s]  [(¬t) (s  r)]
 True

25. Logical Equivalence
Two statements are logically equivalent if they have identical truth tables.

26. Logical Equivalence
A: p  (¬q) and B: ¬(p  q) are identical

27. Tautology
A tautology is a compound statement that is always true, regardless of the truth values assigned to its variables.
Let A: p  (¬q) and B: ¬(p  q)
A  B is always true, so it is a tautology.

28. Contradiction
A Contradiction is a compound statement that is always false.

29. Proposition
A proposition is a synonym for (mathematical) statement.
A proposition is a statement of fact that is either true or false.

30. Proposition Let p and q be propositions.
The proposition “p and q,” or p  q, is true when both p and q are true, and false when one of p or q is false.
p  q is called the conjunction of p and q.

31. Proposition Let p and q be propositions.
The propositions “p or q,” or p  q, is false when p and q are both false and true otherwise.
The proposition p  q is called the disjunction of p and q.

32. Some Basic Logical Equivalences
Idempotence:
(p  q)  p
(p  p)  p

33. Some Basic Logical Equivalences
2. Commutative:
(p  q)  (q  p)
(p  q)  (q  p)

34. Some Basic Logical Equivalences
3. Associativity:
((p  q)  r)  p  (q  r))
((p  q)  r)  (p  (q  r))

35. Some Basic Logical Equivalences
4. Distributivity:
(p  (q  r))  ((p  q)  (p  r))
(p  (q  r))  ((p  q)  (p  r))

36. Some Basic Logical Equivalences
5. Double Negation:
¬(¬p) p
6. De Morgan’s Laws:
¬(p  q)  ((¬p)  (¬q))
¬(p  q)  ((¬p)  (¬q))

37. Some Basic Logical Equivalences
Any two tautologies are logically equivalent.
Any two contradictions are logically equivalent.
Letting 1 denote a tautology and 0 a contradiction.

38. Some Basic Logical Equivalences7.
(p  1) 1
(p  1) p 8.
(p  0) p
(p  0) 0

39. Some Basic Logical Equivalences9.
(p  (¬p)) 1
(p  (¬p)) 0
10.
(¬1)  0
(¬0)  1

40. Some Basic Logical Equivalences
11. (p  q)  [(¬q)  (¬p)]
12. (p  q)  [(¬p)  q]
13. (p  q)  [(p  q)  (q  p)]

41. Show that (¬p) (p q) is a tautology
Using property 12 (p  q)  [(¬p)  q]
[(¬p) (p q)]  [(¬p)  ((¬p)  q)]
 [(¬(¬p)  ((¬p)  q)]
 p  [(¬p)  q]
 [p  (¬p)]  q 1  q 1

42. Verify the first idempotence property
(p  p)  p

43. Verify the second distributive property
(p  (q  r))  ((p  q)  (p  r))

44. Simplify the following statement
[¬(p  q)]  [(¬p)  q]
 [[(¬p)  (¬p)]  [(¬p)  q]]
 by one of the laws of De Morgan using distributive law.

45. Simplify the following statement
 [[(¬p)  (¬p)]  [(¬p)  q]]
 (¬p)  [(¬q)  q]
 (¬p)  1  ¬p
so the given statement is logically equivalent simply to ¬p

46. Theorem 1.2.1
Suppose A and B are logically equivalent statements involving variables p1, p2 …, pn. Suppose that C1, C2, .., Cn are statements.
If, in A and B, we replace p1 by C1, p2 by C2, and so on until we replace pn by Cn, then the resulting statements will still be logically equivalent.

47. Show that
[(p  q)  ((q  (¬r))  (p  r))]  ¬[(¬p) (¬q)]
Using one of the distributive laws, the left-hand side is logically equivalent to
[(p  q)  (q  (¬r))]  [(p  q)  (p  r)]

48. Associativity and idempotence say that the first term here
[(p  q)  ((q  (¬r))  (p  r))]  ¬[(¬p) (¬q)]
Associativity and idempotence say that the first term here
[(p  q)  (q  (¬r))]
Is logically equivalent to
[p  (q  (q  (¬r)))]  [p  ((q  q)  (¬r)))]
 [p  (q  (¬r))]  [(p  q)  (¬r)]

49. The second term [(p  q)  (p  r)] is logically equivalent to
[(p  q)  ((q  (¬r))  (p  r))]  ¬[(¬p) (¬q)]
The second term [(p  q)  (p  r)] is logically equivalent to
[(q  p)  (p  r)]  [q  (p  (p  r))]
 [q  ((p  p)  r)]
 [q  (p  r)]
 [q  (p  r)]  [(p  q)  r]

50. [((q  p)  (¬ r)  ((p  q)  q) ] 
[(p  q)  ((q  (¬r))  (p  r))]  ¬[(¬p) (¬q)]
The expression on he left-hand side of the statement is logically equivalent to
[((q  p)  (¬ r)  ((p  q)  q) ] 
 [(p  q)  0]  p  q
But this is logically equivalent to the right-hand side of the desired statement by double negation and one of the laws of De Morgan.

51. Logical arguments
A theorem is a statement that can be shown to be true (theorems are sometimes called propositions, facts, or results.)
Proof is a sequence of statements that form an argument to demonstrate a theorem.

52. Logical arguments
The statements used in a proof can include axioms or postulates, which are the underlying assumptions about mathematical structures, the hypotheses of the theorem to be proved, and previously proved theorems.

53. Logical arguments
The rules of inference, which are the means used to draw conclusions from other assertions, tie together the steps of a proof.
A lemma (plural lemmas or lemmata) is a simple theorem used in the proof of other theorems. Complicated proofs are usually easier to understand when they are proved using a series of lemmas, where each lemma is proved individually.

54. Logical arguments
A corollary is a proposition that can be established directly from a theorem that has been proved.
A conjecture is a statement whose truth value is unknown. When a proof of a conjecture is found, the conjecture becomes a theorem. Conjectures are shown to be false, so they are not theorem.

55. An Argument
An argument is a finite collection of statements A1, A2, …, An called premises (or hypotheses) followed by a statement B called the conclusion.
Such an argument is valid if, whenever A1, A2, …, An are all true, then B is also true.

56. Show that the argument p  ¬q r  q ______ p  ¬q ¬q is valid r  q
The solution requires the construction of a truth table

57. row 3 is the only row where the premises are all true.
p  ¬q
r  q
______ ¬q
p  ¬q (T)
r  q (T)
r (T)
row 3 is the only row where the premises are all true.

58. p  ¬q
r  q
______ ¬q
The argument can be shown to be valid without the construction of a truth table.
Assume that all premises are true r is true, r  q is also true, q must also be true.
Thus ¬q is false and, because p  (¬q) is true, p is false. Thus ¬p is true as desired.

59. Determine whether the following argument is valid.
If I like biology, then I will study it.
Either I study biology or I fail the course.
___________________________________
If I fail the course, then I do not like biology
Let p, q, and r be the statements.

60. q: “I study biology,” and r: “I fail the course.”
Solution
p: “I like biology,”
q: “I study biology,” and
r: “I fail the course.”
p  q
q  r
_______
r  (¬p)

61. The analysis by means a truth table
5 rows where the premises are all true.

62. Theorem 1.3.2
An argument with premises A1, A2, …, An and conclusion B is valid precisely when the compound statement A1 A2  …  An B is a tautology.

63. Theorem 1.3.3
Substitution: Assume that an argument with premises A1, A2, …, An and conclusion B is valid and that all these statements involves variables p1, p2 …, pn. If p1, p2 …, pn are replaced by statements C1, C2, …, Cn, the resulting argument is still valid.

64. Rules of Inference
Rules of inference provide the justification of the steps used to show that a conclusion follows logically from a set of hypotheses.

65. The tautology (p  (p  q) )  q
Rules of Inference
The tautology (p  (p  q) )  q
is the basis of the rule of inference called modus ponens, or the law of detachment.
This tautology is written in the following way:

66. Rules of Inference

67. Rules of Inference

68. Modus Ponens P
p  q
______ q
The symbol denotes “therefore”
Modus ponens states that if both an implication and its hypothesis are known to be true, then conclusion of this implication is true.

69. “If it snows today, then will go skiing”
Example
“If it snows today, then will go skiing”
The hypothesis, “it is snowing today”, is true,
then by modus ponens, it follows that the conclusion of this implication “we will go skiing,” is true.

70. “If n is greater than 3, then n2 is greater than 9” is true.
Example
“If n is greater than 3, then n2 is greater than 9” is true.
Consequently, if n is greater than 3, then by modus ponens, it follows that n2 is greater than 9.

71. State which rule of inference is the basis of the following argument:
Example
State which rule of inference is the basis of the following argument:
“it is below freezing now. Therefore, it is either below freezing or raining now.”
Soution:
Let p and q be propositions.

72. p: “It is below freezing now,” q: “It is raining now.”
Solution
p: “It is below freezing now,”
q: “It is raining now.”
Then this argument is of the form p
______
 p  q This argument uses the addition rule.

73. State which rule of inference is the basis of the following argument:
Example
State which rule of inference is the basis of the following argument:
“It is below freezing and raining now. Therefore, it is below freezing now.”
Soution:
Let p and q be propositions.

74. p: “It is below freezing now” q: “It is raining now.”
Solution
p: “It is below freezing now”
q: “It is raining now.”
This argument is of the form
p  q
______
 p  This argument uses the simplification rule.

75. Show that the following argument is valid
(p  q)  (s  t)
[¬((¬s)  (¬t))] [(¬r)  q]
_______________________________________________
(p  q) (r  q)

76. One of De Morgan and the principle of double negation tell us that
(p  q)  (s  t)
[¬((¬s)  (¬t))] [(¬r)  q]
___________________________
(p  q) (r  q)
One of De Morgan and the principle of double negation tell us that
[¬((¬s)  (¬t))] 
[(¬(¬s))  (¬(¬t))]
 (s  t)

77. Property 12 of logical equivalence says that [(¬r)  q] (r  q)
(p  q)  (s  t)
[¬((¬s)  (¬t))] [(¬r)  q]
___________________________
(p  q) (r  q)
Property 12 of logical equivalence says that [(¬r)  q] (r  q)
(p  q (s  t)
(s  t) (r  q)
_______________________
(p  q) (r  q)
The chain rule now tells us that our argument is valid.

78. Determine the validity of the following argument
If I study, then I will pass.
If I do not go to a movie, then I will study.
I failed
___________________________________
Therefore, I went to a movie.

79. Let p, q, and r be the statements
p: “I study,”
q: “I pass,” and
r: “I go to a movie.”
p  q
(¬r)  q ¬q
__________ r

80. Let p, q, and r be the statements
The first two premises imply the truth of (¬r)  q by the chain rule.
Since (¬r)  q and ¬q imply ¬ (¬r) by modus tollens
The validity of the argument follows by the principle of double negation: ¬ (¬r)  r
p  q
(¬r)  q ¬q
__________ r

81. Topics covered Truth Tables. The Algebra of Propositions.
Basic logical equivalences
Logical Arguments.
Rules of inference

82. Reference
“Discrete Mathematics with Graph Theory”, Third Edition, E. Goodaire and Michael Parmenter, Pearson Prentice Hall, pp