General Mathematics

1. The Nature of Mathematics

2. Mathematical Reasoning
Part 3

3. ● Mathematical reasoning is the critical skill that
enables us make use of all other mathematical skills.
● Through mathematical reasoning, we are able to
reflect solutions to problems and determine
whether or not they make sense.
Mathematical reasoning

4. ● Inductive reasoning
- It is the process of making general conclusions
based on specific examples.
examples:
- Every object that I release from my hand falls
to the ground. Therefore, the next object I release
from my hand will fall to the ground.
Inductive and deductive
reasoning

5. ● Inductive reasoning
examples:
- Every crow I have ever seen is black.
Therefore, all crows are black.
- Based on the data, the Earth has revolved
around the sun following an elliptical path for millions
of years. Therefore, the Earth will continue to
revolve around the sun in the same manner next
year.
Inductive and deductive
reasoning

6. ● Deductive reasoning
- It is the process of making specific conclusions
based on general principles
examples:
- All men are mortal. I am a man. Therefore, I
am mortal.
(General principle: Modus ponens)
Inductive and deductive
reasoning

7. ● Deductive reasoning
examples:
- Given two supplementary angles with one of
them measuring , the measure of the other angle
angle is .
(General principle: Supplementary angles add up
to .)
Inductive and deductive
reasoning

8. ● Deductive reasoning
examples:
- If , then .
(General principle: If a, b, and c are real numbers
and a = b, then ac = bc)
Inductive and deductive
reasoning

9. ● Comparing these two approaches further, consider
science and mathematics.
- Science is the application of inductive
reasoning to build knowledge based on observable
evidence.
- Every statement in science is considered a
theory. The only way to prove it is to collect more
evidence.
- In inductive reasoning, there is always the
possibility that the future evidence could prove the
statement false.
Inductive versus deductive
reasoning

10. ● Comparing these two approaches further, consider
science and mathematics.
- Mathematics on the other hand, is deductive
reasoning applied to relations among patterns, shapes,
forms, structures, and even changes.
- Deductive reasoning is always valid.
- It makes use of undefined terms, formally
defined terms, axioms, theorems, and rules of
inference.
Inductive versus deductive
reasoning

11. ● A theorem is a statement that can be shown to be
true.
● It is formulated by using a sequence of statements
that form an argument, called proof.
● The rules of inference tie together the steps of a
proof.
Inductive versus deductive
reasoning

12. ● An argument is a collection of propositions where it
is claimed that one of the propositions called the
conclusion follows from the other propositions called
the premises of the argument.
Definition of argument

13. ● An argument in a propositional logic is a sequence of
propositions. All but the final proposition in the
argument are called premises, and the final
proposition is called the conclusion.
● An argument is valid if the truth of all its premises
implies that the conclusion is true
Definition of argument

14. Example:
- Given the following arguments, identify the
premises and the conclusion.
1. Extensive exercise is good for the health.
Good health guarantees clear thinking. So, I
recommend extensive exercise to my students
Definition of argument
Conclusion
Premises

15. Example:
- Given the following arguments, identify the
premises and the conclusion.
2. I believe that Ken is the best prospect for the
highest position in the company. He is very
intelligent and articulate. To this day, he does all his
duties conscientiously. I have not heard of anyone
complain about him since he gets along very well
with his subordinates and colleagues. He has a clear
vision of the direction the company should take. He
is also well respected in the business community.
Definition of argument
Premises
Conclusion

16. ● An argument is valid if:
- the truth of the premises logically guarantees
the truth of the conclusion or
- whenever the premises are all true, the
conclusion is also true.
Definition of a valid argument

17. • Argument form:
.
.
.
_____
Definition of a valid argument
Where:
, … = premises
= conclusion

18. ● Propositional form
Where:
, … = premises
= conclusion
Definition of a valid argument

19. ● An argument is valid if:
- the truth of the premises logically guarantees
the truth of the conclusion or
- whenever the premises are all true, the
conclusion is also true.
Definition of a valid argument

20. ● An argument is said to be
valid if whenever the
premises are all true, the
conclusion is also true.
Thus, the argument :
Definition of a valid argument
.
.
.
_____
is valid if and only if the
propositional form
is a tautology. Otherwise, the
argument is invalid.
(𝒑𝟏 ∧𝒑𝟐∧...∧𝒑𝒏)→𝒒

21. ● Rules of inference
- These are logical forms that are used to
deduce new statements from the statements whose
truth that we already know.
- These rules allow us to make conclusions from
given premises.
Proof of validity

22. ● Rules of inference
- They deal with propositions which can be
inferred from other propositions. These rules consist
of antecedents (premises) and consequents
(conclusion).
- For easy application to arguments, the rules
are expressed in argument form.
Proof of validity

23. ● Rules of inference
- The rules of inference specify conclusions
which can be drawn from assertions known or
assumed to be true. These rules are called by
special names and are stated in argument form.
Proof of validity

24. 1. Addition (Add)
Rules of inference
Illustration:
“It is below freezing
now. Therefore, it is either below
freezing or raining now.”

25. 2. Simplification
(Simp)
Rules of inference
Illustration:
“It is below freezing and
raining now. Therefore, it is
below freezing.”

26. 3. Conjunction (Conj)
Rules of inference
Illustration:
“It is below freezing
now. It is raining now. Therefore,
it is below freezing and raining
now.”

27. 4. Modus ponens
(MP)
Rules of inference
Illustration:
“If you have access to
the network, then you can
change your grade. You have
access to the network.
Therefore, you can change your
grade.”

28. 5. Modus tollens
(MT)
q
Rules of inference
Illustration:
“If you have access to
the network, then you can
change your grade. You have no
access to the network.
Therefore, you cannot change
your grade.”

29. 5. Modus tollens
(MT)
q
Rules of inference
Illustration:
“If you have access to
the network, then you can
change your grade. You have no
access to the network.
Therefore, you cannot change
your grade.”

30. 6. Hypothetical
Syllogism (HS)
Rules of inference
Illustration:
“If it rains today, then
we will stay home. If we stay
home, then we will watch
Netflix. Therefore, if it rains
today, then we will watch
Netflix.”

31. 7. Disjunctive
Syllogism (DS)
Rules of inference
Illustration:
“I go to work or I go to
the movies. I do not go to work.
Therefore, I go to the movies.”

32. 8. Constructive
dilemma (CD)
Rules of inference
Illustration:
“If I study well then I
will get good grades, and if I
finish school then I will be
successful. I study well or I finish
school. Therefore, I will get good
grades or I will be successful.”

33. 9. Destructive
Dilemma (DD)
Rules of inference
Illustration:
“If I go shopping today
then I will go to the party
tonight, and if I get hungry then
I will cook dinner. I will not go to
the party tonight or I will not
cook dinner. Therefore, I will not
go shopping today or I do not
get hungry.”

34. ● The corresponding propositional forms of these
rules of inference can be shown as tautologies. As an
example, let us consider the truth table of modus
tollens
Rules of inference
q [( 𝒑 → 𝒒) ∧ ¬𝒒 ] →¬ 𝒑
Argument form
Propositional form

35. ● The corresponding propositional forms of these
rules of inference can be shown as tautologies. As an
example, let us consider the truth table of modus
tollens :
Rules of inference
[( 𝒑→𝒒) ∧¬𝒒]→¬ 𝒑
T T T T T F F T F
T F T F F F T T F
F T F T T F F T T
F F F T F T T T T
Tautology

36. ● At this point, we will accept all other rules of
inference without verification. We claim that all
these rules are valid arguments.
● The proof that we employ using the definition of
valid arguments is what we call the direct proof.
Direct proof

37. Examples:
For each of the following arguments, construct a
formal proof of validity. State the justification (rule of
inference) for each of the line that is not a premise.
1.
_______________
Direct proof
3 & 5 - Conj
3 - Add
2 & 4 - MP
1 - Simp

38. Examples:
2.
_______________
Direct proof
2 & 4 - CD
5 & 3 - DS
1 - Simp

39. Examples:
3.
_______________
Direct proof
5 & 3 - Conj
1 & 2 - HS
6 & 4 - CD

40. Examples:
4.
_______________
Direct proof
1 & 5 - Conj
2 & 4 - MP
3 & 4 - DS
6 & 7 - CD

41. Examples:
5. Show that the following is a valid argument.
= John is optimistic
= John is busy
= John will play the lottery
= John will visit the casino
= John is broke
IF John is not optimistic and John is busy
AND If John plays the lottery then John is
optimistic
AND If John does not play the lottery, then he
will visit the casino
AND If John visits the casino, then he will be
broke
THEN John is broke
Direct proof

42. Examples:
5. Show that the following is a valid argument.
Direct proof
IF John is not optimistic and
John is busy.
()
AND If John plays the lottery
then John is optimistic.
()
AND If John does not play the lottery,
then he will visit the casino.
()
AND If John visits the casino,
then he will be broke.
()
THEN John is broke.
()
____________
7.
1 - Simp
2 & 5 - MT
3 & 6 - MP
4 & 7 - MP

43. ● The invalidity of an argument may be verified by
showing that its propositional form is not a
tautology.
● Since the propositional form of an argument is an
implication, then we should be able to show an
instance when the premise is true but the conclusion
is false.
Proof of invalidity of an
argument

44. ● We do not have to construct the whole truth table
for the propositional form to do this. All we have to
do is find the combination of values that makes the
propositional form of the argument false
* An implication is false when the premise is true
but the conclusion is false.
Proof of invalidity of an
argument
T F

45. ● The simplified process of constructing a truth table
is called the shortened truth table method of
showing the invalidity of argument.
● One counterexample which gives the truth values of
the propositions in the argument that make all the
premises true and the conclusion false is enough to
disprove the validity of the argument.
Proof of invalidity of an
argument
T F
T T T F

46. Examples:
Prove the invalidity of the following argument.
1.
_______
Proof of invalidity of an
argument
 transform to propositional form
T F
F
F
F
𝑨=𝐅 , 𝑩=𝐓 , 𝑪=𝐅 , 𝑫=𝐅
F F
T
T
T T T F
F

47. Examples:
Use the shortened truth table to prove the invalidity
of the following argument.
2. )
___________
Proof of invalidity of an
argument
F
T
T F F
F
F
F
T
𝑬=𝐓, 𝑭 =𝐓, 𝑮=𝐅 , 𝑯=𝐅 , 𝑰=𝐅
T F
T T T F

48. Nocon, R. & Nocon E. (2018). Mathematics for the modern
world.
Aufmann, R., Lockwood, J., Nation, R., Clegg, D., Epp, S., Abad
Jr., E. (2018). Mathematics in the modern world (Philippine
edition). Rex Book Store, Inc.
Rosen, K. (2012). Discrete mathematics and its applications
(7th
Edition). McGraw Hill.
Johnsonbaugh, R. (2005). Discrete mathematics (6th
International Edition). Pearson Education, Inc.
References