math as a language

1. MATHEMATICS AS A
LANGUAGE
Mathematics in the Modern World
MATH 100 AY 2023-2024

2. ANSWER:
WHAT’S NEXT?

3. 1 ∗ 1 = 1
11 ∗ 11 = 121
111 ∗ 111 = 12,321
1,111 ∗ 1,111 = 1,234,321
11,111 ∗ 11,111 = 123,454,321
111,111 ∗ 111,111 = ? ? ? ?
WHAT’S NEXT?
ANSWER
: 12,345,654,321

4. WHAT’S NEXT?
1
3
15
7
31
ANSWER
:
1 = 21
− 1
3 = 22
− 1
7 = 23
− 1
15 = 24
− 1
31 = 25
− 1
IS THIS THE
ONLY EXPLANATION?

5. WHAT’S NEXT?
ANSWER
:

6. WHAT’S THE
PATTERN IN
THE GIVEN
SET OF
EQUATIONS?

7. LETS
BEGIN!

8. How did you learn
your native
language?
(Filipino/Chinese
English/Japanese
Korean…)

9. MATHEMATICS AS LANGUAGE
A LANGUAGE is a systematic means of communicating
by the use of sound or conventional symbols. It is the
code we all use to express ourselves and
communicate to others.

10. Components of a language:
Vocabulary of symbols or words
Grammar or rules of how these symbols are used
Community of people who use and understand
these symbols
Range of meanings that can be communicated
with these symbols.

11. • ENGLISH LANGUAGE
• Symbols: English Letters
• Vowels and Consonants
• Words
• Phrases
• Sentences
MATHEMATICAL LANGUAGE
(Algebra)
Symbols: English Letters/ Arabic
Numerals
Variables and Constants
Term
Algebraic Expressions
Mathematical statements:
Equations, Inequalities, etc

12. ELEMENTS OF THE MATHEMATICAL
LANGUAGE
0123456789
±×÷ ∞ =≠ ~ <≥≤∓≅
≡ ∀
4
∪ ∩ ∅ % ∃ ∄ ∈
∋ 𝛼 𝛽 𝛾 𝛿 𝜀 𝜖 𝜃 𝜗 𝜋 𝜇 𝜌 𝜎 𝜏 𝜑 𝜔

13. PROPOSITIONAL CALCULUS
calculus?

14. PROPOSITIONAL CALCULUS
A proposition is a complete
declarative sentence that is either
TRUE or FALSE, but not both.

15. EXAMPLES
1) Manila is the capital of the Philippines.
2) Shanghai is the capital of China.
3) Today is a Tuesday.
4) 1+1=2
5) 2+2=5

16. EXERCISES
Identify if the following
sentences are propositions.
1) Is it time already?
2) Pay attention to this.
3) 𝑥 + 1 = 2
4) 𝑥 + 𝑦 = 𝑧

17. Propositions are usually
denoted by capital
letters of the English
alphabet.
(But most of the time we
use P,Q,R,S and T)

18. If a proposition 𝑃 is true, its truth value
is 𝑡𝑟𝑢𝑒, and is usually denoted by 𝑇.
If it is false, its truth value if
𝑓𝑎𝑙𝑠𝑒 denoted by 𝐹.

19. EXAMPLES
𝑴: The class of Mr. Garcia is very attentive.
𝑵: Students of Dr. Nocon are attentive in class.
𝑺: Students in this class are all science students.

20. CONNECTIVES AND COMPOUND
PROPOSITIONS
A propositional connective is an operation
that combines two propositions to yield a new
proposition whose truth value depends only on
the truth values of the two original
propositions.

21. CONNECTIVES AND COMPOUND
PROPOSITIONS
Combinations of propositions using
propositional connectives are called
compound proposition.

22. PROPOSITIONAL CONNECTIVES
∧ conjunction (and)
∨ disjunction (or )
⨁ 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒 𝑜𝑟
⇒ implication (implies)
⇔ 𝑏𝑖𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝒊𝒇 𝒂𝒏𝒅 𝒐𝒏𝒍𝒚 𝒊𝒇
¬ 𝑛𝑒𝑔𝑎𝑡𝑖𝑜𝑛 (𝒏𝒐𝒕)

23. EXAMPLES
𝑷 : Alyssa is sleeping.
𝑸 : Matthew is noisy.
𝑹 : Kyla is late in her class.
What does the following stand for?
1) 𝑃 ∧ 𝑅
2) ¬𝑄 ⇒ 𝑃
3) 𝑄 ∨ 𝑅

24. If propositions have their
truth values?
What the truth value of a
compound proposition?

25. NEGATION OF A PROPOSITION
𝑷 ¬𝑷
T F
F T
The negation of a proposition 𝑃 is denoted
by ¬𝑃 and is read as “not 𝑃”.
Truth Table

26. EXAMPLE
1. 𝑃: It will rain today.
¬𝑃: It will not rain today.
2. 𝑄: Samantha is hardworking.
¬𝑄: Samantha is not hardworking.
3. 𝑅: You will pass this course.
¬𝑅: You will not pass this course.

27. CONJUNCTION OF PROPOSITIONS
The proposition “𝑃 and 𝑄”, denoted by
𝑷 ∧ 𝑸,
is called the conjunction of 𝑃 and 𝑄.

28. TRUTH VALUE OF 𝑷 ∧ 𝑸
𝑷 𝑸 𝑷 ∧ 𝑸
T T T
T F F
F T F
F F F

29. EXAMPLE
1. 𝑃: Althea is beautiful.
𝑄: Lance is strong.
𝑃 ∧ 𝑄: Althea is beautiful and Lance is strong.
2. 𝑆: The stock exchange is down.
𝑇: The stock exchange will continue to decrease.
𝑆 ∧ 𝑇: The stock exchange is down and it will continue to
decrease.

30. DISJUNCTION: INCLUSIVE “OR”
The proposition “𝑃 or 𝑄”, denoted by
𝑷 ∨ 𝑸,
is called the disjunction of 𝑃 or 𝑄. This is also
referred to as the inclusive “or”.

31. TRUTH VALUE OF 𝑷 ∨ 𝑸
𝑷 𝑸 𝑷 ∨ 𝑸
T T T
T F T
F T T
F F F

32. EXAMPLE: INCLUSIVE “OR”
1. 𝑃: This lesson is interesting.
𝑄: The lesson is easy.
𝑃 ∨ 𝑄: This lesson is interesting or it is easy.
2. 𝑆: I want to take a diet.
𝑇: The food is irresistible.
𝑆 ∨ 𝑇: I want to take a diet or the food is irresistible.

33. DISJUNCTION: EXCLUSIVE “OR”
The proposition “𝑃 or 𝑄 but not both”, denoted
by
𝑷⨁𝑸.
This is also referred to as the “exclusive or”.

34. TRUTH VALUE OF 𝑷⨁𝑸
𝑷 𝑸 𝑷⨁𝑸
T T F
T F T
F T T
F F F

35. IMPLICATIONS OR CONDITIONALS
The proposition “If 𝑷, then 𝑸”, denoted by
𝑷 ⇒ 𝑸
is called an implication or a conditional.
Equivalent propositions: “𝑷 only if 𝑸”, “𝑸 follows
from 𝑷”, “𝑷 is a sufficient condition for 𝑸”, “𝑸
whenever 𝑷”

36. THE COMPOUND PROPOSITION
𝑷 ⟹ 𝑸
Also called the conditional statement.
𝑷 ⟹ 𝑸
Hypothesis
Antecedent
Premise
Conclusion
Consequence

37. THE MANY NAMES OF 𝑷 ⟹ 𝑸

38. TRUTH VALUE OF 𝑷 ⟹ 𝑸
𝑷 𝑸 𝑷 ⟹ 𝑸
T T T
T F F
F T T
F F T

39. EXAMPLE
𝑷: It is raining very hard today.
𝑸: Classes are suspended.
𝑷 ⇒ 𝑸: If it is raining very hard today,
then classes are suspended.

40. TOO MANY TABLES….

41. PROPOSITIONS RELATED TO
𝑷 ⟹ 𝑸
CONVERSE CONTRAPOSITVE INVERS
E

42. RELATED IMPLICATION: CONVERSE
The converse of the proposition
“If 𝑃, then 𝑄”
is the proposition
“If 𝑄, then 𝑃”.
In symbols, the converse of
𝑃 ⇒ 𝑄 is 𝑄 ⇒ 𝑃.

43. The converse of the proposition 𝑃 ⇒ 𝑄
“If it is raining very hard today, then classes are
suspended.”
is 𝑄 ⇒ 𝑃 and is stated as
“If classes are suspended, then it is raining very hard
today.”
EXAMPLE

44. RELATED IMPLICATION:
CONTRAPOSITIVE
The contrapositive of the proposition
“If 𝑃, then 𝑄”
is the proposition
“If not 𝑄, then not 𝑃”.
In symbols, the contrapositive of
𝑷 ⇒ 𝑸 is ¬𝑸 ⇒ ¬𝑷.

45. The contrapositive of the proposition 𝑃 ⇒ 𝑄:
“If it is raining very hard today, then classes are
suspended.”
is the proposition ¬𝑄 ⇒ ¬𝑃:
“If classes are not suspended, then it is not raining
very hard today.”
EXAMPLE

46. RELATED IMPLICATION: INVERSE
The inverse of the proposition
“If 𝑃, then 𝑄”
is the proposition
“If not 𝑃, then not 𝑄”.
In symbols, the inverse of
𝑃 ⇒ 𝑄 is ¬𝑃 ⇒ ¬𝑄.

47. The inverse of the proposition 𝑃 ⇒ 𝑄:
“If it is raining very hard today, then classes are
suspended.”
is the proposition ¬𝑄 ⇒ ¬𝑃:
“If it is not raining very hard today, then classes are
not suspended.”
EXAMPLE

48. BICONDITIONALS
The proposition
“𝑷 if and only if 𝑸”,
denoted by
𝑷 ⇔ 𝑸
is called a biconditional.
Equivalent propositions: “𝑃 is equivalent to 𝑄”, “𝑃 is a
necessary and sufficient condition for 𝑄”

49. SUMMARY OF TRUTH TABLES
𝑷 𝑸 𝑷 ∧ 𝑸 𝑷 ∨ 𝑸 𝑷⨁𝑸 𝑷 ⟹ 𝑸 𝑷 ⟺ 𝑸 ¬𝑷 ¬𝑸
T T T T F T T F F
T F F T T F F F T
F T F T T T F T F
F F F F F T T T T

50. How can you determine
the number of rows in a
truth table?
Counting
Technique!!
Statistics
ALERT!!

51. If there are N propositions then the number
of rows is
𝟐𝑵
𝑷 𝑸 𝑷 ⟹ 𝑸
T T T
T F F
F T T
F F T
𝑷 ¬𝑷
T F
F T

52. TIPS IN CONSTRUCTING
TRUTH TABLES
See the
pattern?

53. EXERCISES
Let 𝑃 and 𝑄 be the propositions
𝑷 “I am a Math major.”
𝑸 “I love Mathematics.”

54. WHAT'S THE TRUTH TABLE FOR
¬𝑷 ∨ 𝑸
𝑷 𝑸 ¬𝑷 ¬𝑷 ∨ 𝑸

55. TAUTOLOGY, CONTRADICTION AND
CONTINGENCY
•A compound proposition that is
ALWAYS true – TAUTOLOGY
ALWAYS false – CONTRADICTION
SOMETIMES true – CONTINGENCY

56. EXAMPLE
•Consider the following propositions and
determine if it is a tautology, contradiction or
a contingency.
1. 𝑝 ∨ ¬𝑝
2. 𝑝 ∧ ¬𝑝

57. LOGICALLY EQUIVALENT
•“Two propositions 𝑝, 𝑞 are logically
equivalent if 𝑝 ⟺ 𝑞 is a tautology.”
Show that ¬(𝑝 ∨ 𝑞) and ¬𝑝 ∧ ¬𝑞 are
logically equivalent.

58. SOME LOGICAL EQUIVALENCES
Logical Equivalence Name
𝑝 ∧ 𝑻 ⟺ 𝑝
IDENTITY LAWS
𝑝 ∨ 𝑭 ⟺ 𝑝
𝑝 ∨ 𝑻 ⟺ 𝑻
DOMINATION LAWS
𝑝 ∧ 𝑭 ⟺ 𝑭
𝑝 ∧ 𝑝 ⟺ 𝑝 IDEMPOTENT LAWS
¬(¬𝑝) ⟺ 𝑝 DOUBLE NEGATION

59. SOME LOGICAL EQUIVALENCES
Logical Equivalence Name
𝑝 ∧ 𝑞 ⟺ 𝑞 ∧ 𝑝
COMMUTATIVE LAWS
𝑝 ∨ 𝑞 ⟺ 𝑞 ∨ 𝑝
(𝑝 ∧ 𝑞) ∧ 𝑟 ⟺ 𝑝 ∧ (𝑞 ∧ 𝑟)
ASSOCIATIVE LAWS
(𝑝 ∨ 𝑞) ∨ 𝑟 ⟺ 𝑝 ∨ (𝑞 ∨ 𝑟)
𝑝 ∨ (𝑞 ∧ 𝑟) ⟺ (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟)
DISTRIBUTIVE LAWS
𝑝 ∧ (𝑞 ∨ 𝑟) ⟺ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟)
¬(𝑝 ∧ 𝑞) ⟺ ¬𝑝 ∨ ¬𝑞
DE MORGAN’S LAW
¬(𝑝 ∨ 𝑞) ⟺ ¬𝑝 ∧ ¬𝑞

60. ON IMPLICATIONS
•𝑝 ⇒ 𝑞
Hypothesis Conclusion
CONVERSE:
𝑞 ⇒ 𝑝
INVERSE:
¬𝑝 ⇒ ¬𝑞
CONTRAPOSITIVE:
¬𝑞 ⇒ ¬𝑝
“An implication is always logically equivalent to
its own contrapositive.”

61. WHEN IS MATHEMATICAL
REASONING CORRECT?
Deductiv
e
Reasonin
g
Inductive
Reasonin
g

62. MATHEMATICAL JARGONS
THEOREM
PROOF
AXIOM
RULES OF INFERENCE

63. SOME RULES OF INFERENCE
Rule of
Inference
Name Rule of
Inference
Name
𝑝
_______
∴ 𝑝 ∨ 𝑞
ADDITION
¬𝑞
𝑝 ⇒ 𝑞
______
∴ ¬𝑝
MODUS TOLLENS
(the mode of denying)
𝑝 ∧ 𝑞
_______
∴ 𝑝
SIMPLIFICATION
𝑝 ⇒ 𝑞
𝑞 ⇒ 𝑟
________
∴ 𝑝 ⇒ 𝑟
HYPOTHETICAL
SYLLOGISM
𝑝
𝑞
______
∴ 𝑝 ∨ 𝑞
CONJUNCTION
𝑝 ∨ 𝑞
¬𝑝
_______
∴ 𝑞
DISJUNCTIVE
SYLLOGISM
𝑝
𝑝 ⇒ 𝑞
______
∴ 𝑞
MODUS PONENS
(the mode of
affirming)

64. EXAMPLES
Identify the rules of inference used in each of the
following arguments.
1. Anna is a human resource management major.
Therefore, Anna is either a human resource
management major or a computer applications major.
2. If you have a current network password, then you can
log on to the network. You have a current network
password. Therefore, you can log on to the network.

65. EXAMPLES
3. If you have a current network password, then you
can log on to the network. You can’t log on to the
network. Therefore, you don’t have a current
network password.
4. If I go swimming, the I will stay in the sun for an
hour. If I stay in the sun for an hour, then I will get
sunburn. Therefore, if I go swimming , then I will
get sunburn.

66. TYPES OF FALLACIES
Fallacy of affirming conclusion
Fallacy of denying the hypothesis
Begging the question or circular reasoning

67. EXAMPLE
• If you do every problem in a math book, then you
will learn mathematics. You learned mathematics.
Therefore, you did every problem in a math book.
Fallacy of affirming
the conclusion!

68. EXAMPLE
•If you do every problem in a math book, then you
will learn mathematics. You did not do every
problem in the math book. Therefore, you did not
learn mathematics.
Fallacy of denying the
hypothesis!

69. EXERCISES
1. Answer the following exercises found on page
15 of the book: #’s 8,9,12,16 and 18.
2. Answer the following exercises found on page
22 of the book: #’s 1 and 2.