The rules of logic give a precise meaning to mathematical statements. These rules are used to distinguish between valid and invalid mathematical arguments. We begin our study of discrete mathematics with an introduction to logic.

1. MA.WENNILOU
Z.PORAZO
DISCRETE MATHEMATICS LECTURE 1
PROPOSITIONAL
LOGIC
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2. LOGIC
Definition
the basis of all mathematical reasoning and all of
automated reasoning.
Rules of logic are used to distinguish between
valid and invalid mathematical arguments.
discrete mathematics

3. propositions
Definition
basic building blocks of logic.
a declarative sentence that is either true or false
but not both.
discrete mathematics

4. WHICH OF THESE
ARE
PREPOSITIONS?
Tokyo is the
capital of Japan
1 + 1 = 2
2 + 5 = 11
discrete mathematics
What time is it?
Read this carefully.
x + y = 12

5. • WE USE LETTERS TO DENOTE PROPOSITIONAL
VARIABLES (OR STATEMENT VARIABLES), THAT
IS, VARIABLES THAT REPRESENT
PROPOSITIONS, JUST AS LETTER ARE USED TO
DENOTE NUMERICAL VARIABLES.
• It uses the conventional letters such as p,q,r,s,...
• Truth values is denoted by T, if it's a true proposition, and
how do we represent
prepositions?
discrete mathematics

6. propositional
logic
• Area of logic that deals with prepositions
• First developed systematically by Greek
philosopher Aristotle more than 2300 years ago.
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7. compound
propositions
• New propositions formed from existing propositions
using logical operators.
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8. discrete mathematics
Let p be a proposition. The
negation of p, denoted by ¬p, is the
statement
"It is not the case that p"
The truth value of the negation of p,
¬p, is the opposite of the truth value
of p.
NEGATION
Definition 1

9. EXAMPLE 1
FIND THE NEGATION OF THE PROPOSITION
Michelle's PC
runs Linux.
SOLUTION
DECEMBER 2020
It's not the case that
Michelle's Pc runs Linux.
Michelle's PC doesn't run
Linux.
PROBLEM

10. EXAMPLE 2
FIND THE NEGATION OF THE PROPOSITION
Vanessa's smartphone has at
least 32GB of memory.
SOLUTION
DECEMBER 2020
It's not the case that Vanessa's
smartphone has at least 33GB of
memory.
PROBLEM
Vanessa's smartphone doesn't have
at least 32GB of memory.

11. DECEMBER 2020
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Truth table for the negation of
proposition of p
Somethingtothinkabout

12. Let p and q be propositions. The
conjunction of p and q, denoted by
p ∧ q ,is the proposition.
"p and q"
The conjunction of p and q is true
when both p and q are true and
false if otherwise.
conjunction
DECEMBER 2020
Definition 2

13. SAMPLE PROBLEM
FIND THE CONJUNCTION OF THE PROPOSITIONS P AND Q
WHERE P IS THE PROPOSITION:
Rachelle's PC has more than 50GB free hard disk space.
DECEMBER 2020
AND Q IS THE PROPOSITION:
The processor in Rachelle's PC runs faster than 2 GHz.

14. SOLUTION
THE CONJUNCTION OF THESE PREPOSITIONS, P ∧ Q IS
THE PREPOSITION:
Rachelle's PC has more than 50GB free hard disk space AND the processor
in Rachelle's PC runs faster than 2 GHz.
DECEMBER 2020

15. Truth table for the conjunction of two
propositions.
DECEMBER 2020
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Somethingtothinkabout

16. Let p and q be propositions. The
disjunction of p and q, denoted by
p ∨ q ,is the proposition.
"p or q"
The disjunction of p ∨ q is
false when both p and q are false
and is true if otherwise.
disjunction
DEMCEMBER 2020
Definition 3

17. SAMPLE PROBLEM
FIND THE DISJUNCTION OF THE PROPOSITIONS P AND Q
WHERE P IS THE PROPOSITION:
Rachelle's PC has more than 50GB free hard disk space.
DECEMBER 2020
AND Q IS THE PROPOSITION:
The processor in Rachelle's PC runs faster than 2 GHz.

18. SOLUTION
THE DISJUNCTION OF THESE PREPOSITIONS, P ∨ Q IS
THE PREPOSITION:
Rachelle's PC has more than 50GB free hard disk space OR the processor
in Rachelle's PC runs faster than 2 GHz.
DECEMBER 2020

19. Truth table for the disjunction of two
propositions.
DECEMBER 2020
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Somethingtothinkabout

20. Let p and q be 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 and is false otherwise.
exclusive
or
DECEMBER 2020
Definition 4

21. SAMPLE PROBLEM
FIND THE EXCLUSIVE OR OF THE PROPOSITIONS P AND Q
WHERE P IS THE PROPOSITION:
AND Q IS THE PROPOSITION:
Rachelle's PC has more than 50GB free hard disk space.
DECEMBER 2020
The processor in Rachelle's PC runs faster than 2 GHz.

22. SOLUTION
THE EXCLUSIVE OR OR P AND Q,(P ⊕ Q) IS THE
PROPOSITION:
Rachelle's PC has at least 50GB free hard disk space , OR the processor
in Rachelle's PC runs faster than 2 GHz, but not both.
DECEMBER 2020

23. Truth table for the Exclusive OR of
two propositions.
DECEMBER 2020
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Somethingtothinkabout

24. Let p and q be propositions. The
conditional statement p --> q is the
proposition, "if p, then q".
The conditional statement p --> q is
false when p is true and q is false,
and true otherwise.
CONDITIONAL
STATEMENT
DECEMBER 2020
Definition 5
In the conditional statement p --> q, p is called the
hypothesis (or the antecedent or premise) and q is
called the conclusion (or the consequence)

25. "if p, then q"
p → q
august 2020 discrete mathematics
"if p, q"
"q if p"
"q when p"
"q follows from p"
"a necessary condition for p is q"
"a sufficient condition for q is p"
"p implies q"
"p only if q"
"q whenever p"
"q is necessary for p"

26. SAMPLE PROBLEM 1
LET P BE THE STATEMENT:
AND Q THE STATEMENT:
"Joy learns Discrete Mathematics."
DECEMBER 2020
"Joy will find a good job."
EXPRESS THE STATEMENT P → Q INTO ENGLISH

27. SOLUTION
P → Q
"If Joy learns Discrete Mathematics, then she
will find a good job."
DECEMBER 2020

28. SAMPLE PROBLEM 2
LET P BE THE STATEMENT:
AND Q THE STATEMENT:
"I am elected as the town Mayor."
DECEMBER 2020
"I will lower the taxes and tuition fees."
EXPRESS THE STATEMENT P → Q INTO ENGLISH

29. SOLUTION 2
P → Q
"If I am elected as the town Mayor, then I will
lower the taxes and tuition fees."
DECEMBER 2020

30. ANOTHER EXAMPLE
P → Q
"If you get 100 on the exam, then I will give you
1.0 grade."
DECEMBER 2020

31. Truth table for the Conditional
Statement p → q.
DECEMBER 2020
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Somethingtothinkabout

32. CONVERSE,
CONTRAPOSITIVE,
INVERSE
discrete mathematics

33. If you pass the exam, then you
will graduate on time.
converse
If you graduate on time, then you passed
the exam.
contrapositive
inverse
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Proposition:
p → q
If you do not graduate on time, then you
did not pass the exam.
If you don't pass the exam, then you don't
graduate on time.

34. EXPLAINATION
The contrapositive, ¬ q → ¬ p of an implication p → q has the same truth value as p →
q. To see this, note that the contrapositive is false only when ¬ p is false and ¬ q is
true, that is, only when p is true and q is false. On the other hand, neither the
converse, q → p nor the inverse, ¬ p → ¬q has the same truth value as p → q for all
possible truth values of p and q. To see this, note that when p is true and q is false
,the original implication is false, but the converse and inverse are both true. When
two compound propositions always have the same truth value we call them
equivalent, so that an implication and its contrapositive are equivalent. The converse
and inverse of an implication are also equivalent, as the reader can verify.
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35. EXAMPLE: What are the contrapositive, the converse,
and the inverse of the implication?
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“The home team wins whenever it is
raining.”?

36. Solution: Because q whenever p is one of the ways
to express the implication , the original statement
can be written as
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Consequently, the contrapositive of this implication
is:
“If it is raining, then the home team wins.”
“If the home team does not win, then it is not
raining.”

37. The converse is:
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The inverse is:
“If the home team wins, then it is raining.”
“If it is not raining, then the home team does not
win.”

38. Note:
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Only the contrapositive is equivalent
to the original statement.

39. bic nditionals
DECEMBER 2020
discrete mathematics
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40. • Let p and q be propositions. The biconditional statement
p ↔ q, is the proposition "p if and only if q."
• The biconditional statement p ↔ q is true when p and q
has the same truth vales, and false otherwise.
Definition 6
DISCRETE MATHEMATICS
• Biconditional statements are also called bi-implications.
BICONDITIONALS

41. WHAT ARE THE
WAYS TO EXPRESS
P ↔ Q?
"P is necessary and
sufficient for Q"
"If p and q, and
conversely"
"p iff q

42. SAMPLE PROBLEM 1
LET P BE THE STATEMENT:
AND Q THE STATEMENT:
"You can take the flight."
DECEMBER 2020
"You buy a ticket."
EXPRESS THE STATEMENT P ↔ Q USING IFF

43. SOLUTION
THEN THE P ↔ Q STATEMENT:
"You can take the flight, if and only if you buy a
ticket."
DECEMBER 2020

44. Truth table for the Biconditional
Statement p ↔ q.
DECEMBER 2020
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Somethingtothinkabout
Note :
p ↔ q has exactly the same
truth value as...
(p → q) ∧ (q → p)

45. PRECEDENCE OF LOGICAL OPERATORS
discrete mathematics
1
2
3
4
5
∧
∨
→
↔

46. SAMPLE EXERCISE

47. CONSTRUCT THE TRUTH TABLE OF THE COMPOUND
PROPOSITION
exercise 1
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discrete mathematics
(p ∨ ¬q ) → (p ∧ q)

48. solution
exercise 1
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discrete mathematics
(p ∨ ¬q ) → (p ∧ q)