1. Mathematics as a language
Mathematics in the Modern World
GEMATMW Term 2 AY
Point out the following ideas:
Many results in mathematics came about as generalizations of patterns and shape.
Studying patterns allows us to observe, hypothesize, discover and create.
The way of doing mathematics has evolved from just perfroming calculations or making deductions from patterns , testing conjectures and estimating results.
Mathematics has become a diverse discipline dealing with data, measurements and observation from the sciences as well as working with mathematical models of narutal phenomena, human behavior and social systems.
All these tells us that for all these aspects to advance and progress with the help of mathematics, a good grasp of the mathematical language is necessary.
Ask the students what is the language of mathematics.

3. Wait! Before we talk about “MATH”
Lets PLAY A GAME!

4. …just a game of what’s next!!!
Ooopss…
…just a game of what’s next!!!
Mention that we see these kinds of logic questions in entrance exams? Or in some IQ Tests.

5. What’s next?
ANSWER:
Mention that we see these kinds of logic questions in entrance exams? Or in some IQ Tests.

6. What’s next?
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= ????
ANSWER:
12,345,654,321

7. 1 3 7 15 31 What’s next? Is this the only explanation? 1= 2 1 −1
ANSWER:
1= 2 1 −1 3
3= 2 2 −1 7
7= 2 3 −1 15
15= 2 4 −1 31
31= 2 5 −1
Is this the
only explanation?

8. What’s next?
ANSWER:

9. What’s the pattern in the given set of equations?

10. Lets Begin!
Now that your brain muscles are all warmed up!!

11. How did you learn your native language?
(Filipino/Chinese
English/Japanese
Korean…)
Ask the class who knows the following languages? Chinese/Japanese/Korean? Ask them how they learned these languages?

12. 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.
Point out that all these components are found in mathematics

13. 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.

14. Symbols: English Letters Vowels and Consonants
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
Emphasize the comparison between the English Language and Mathematical Language used in Algebra

15. Elements of the mathematical language
±×÷∞=≠~<≥≤∓≅ ≡∀ 4 ∪ ∩ ∅ % ∃ ∄ ∈ ∋𝛼 𝛽 𝛾 𝛿 𝜀 𝜖 𝜃 𝜗 𝜋 𝜇 𝜌 𝜎 𝜏 𝜑 𝜔
Just like any other language, mathematics has nouns, pronouns, verbs and sentences. It has its own vocabulary, grammar, syntax, word order, synonyms, negations, sentence structure etc..

16. Propositional calculus
Mention that the term calculus just means – “a particular method or system of calculation or reasoning”

17. Propositional calculus
A proposition is a complete declarative sentence that is either TRUE or FALSE, but not both.
Example:
There are 4 sections of GEMATMW attending in today’s Master class.
Ask the each section to provide an example of a proposition.

18. EXAMPLES Manila is the capital of the Philippines.
Shanghai is the capital of China.
Today is a Tuesday.
1+1=2
2+2=5
Propositions 1) and 4) are TRUE while 2, 3 and 5 are FALSE.

19. Exercises Identify if the following sentences are propositions.
Is it time already?
Pay attention to this.
𝑥+1=2
𝑥+𝑦=𝑧
1) and 2) are NOT propositions since they are not declarative sentences or statements. While 3) and 4) are not propositions since they are either true or false depending on the values fo the variables.

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

21. 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.
Mention that in Propositional calculus / Mathematical reasoning, Propositions are just like the alphabet of the English language.

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

23. 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.

24. Connectives and compound propositions
Combinations of propositions using propositional connectives are called compound proposition.

25. Propositional connectives
∧ conjunction (and) ∨ disjunction (or ) ⨁ 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒 𝑜𝑟 ⇒ implication (implies) ⇔ 𝑏𝑖𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝒊𝒇 𝒂𝒏𝒅 𝒐𝒏𝒍𝒚 𝒊𝒇 ¬ 𝑛𝑒𝑔𝑎𝑡𝑖𝑜𝑛 (𝒏𝒐𝒕)
Point out that there are other connectives will not be discussed that they can encounter when they read some specialized books in computer science, engineering and advanced mathematics books.

26. Examples
𝑷 : Alyssa is sleeping. 𝑸 : Matthew is noisy. 𝑹 : Kyla is late in her class.
What does the following stand for?
𝑃∧𝑅
¬𝑄⇒𝑃
𝑄∨𝑅

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

28. Negation of a proposition
The negation of a proposition 𝑃 is denoted by ¬𝑃 and is read as “not 𝑃”.
Truth Table 𝑷 ¬𝑷 T F

29. Example 𝑃: It will rain today. ¬𝑃: It will not rain today.
𝑄: Samantha is hardworking.
¬𝑄: Samantha is not hardworking.
𝑅: You will pass this course.
¬𝑅: You will not pass this course.

30. Conjunction of Propositions
The proposition “𝑃 and 𝑄”, denoted by
𝑷∧𝑸,
is called the conjunction of 𝑃 and 𝑄.

31. Truth value of 𝑷∧𝑸 𝑷 𝑸 𝑷∧𝑸 T F
The conjunction of two propositions is TRUE only if both propositons are TRUE otherwise the conjunction is FALSE

32. Example 𝑃: Althea is beautiful. 𝑄: Lance is strong.
𝑃∧𝑄: Althea is beautiful and Lance is strong.
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.

33. 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”.

34. Truth value of 𝑷∨𝑸 𝑷 𝑸 𝑷∨𝑸 T F
The disjunction of two proposition is FALSE if both are FALSE and TRUE otherwise.

35. Example: Inclusive “or”
𝑃: This lesson is interesting.
𝑄: The lesson is easy.
𝑃∨𝑄: This lesson is interesting or it is easy.
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.

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

37. Truth value of 𝑷⨁𝑸 𝑷 𝑸 𝑷⨁𝑸 T F
The exclusive or of two proposition is TRUE if the two propositions have different truth values.

38. 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”

39. The compound proposition 𝑷⟹𝑸
Also called the conditional statement.
𝑷⟹𝑸
Hypothesis
Antecedent
Premise
Conclusion
Consequence

40. the many names of 𝑷⟹𝑸

41. Truth value of 𝑷⟹𝑸 𝑷 𝑸 𝑷⟹𝑸 T F
The implication 𝑷⟹𝑸 will only be FALSE in the case that the conclusion is false by the premise is true. And in all other cases, the implication is TRUE.

42. 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.

43. Too many tables….

44. Propositions related to
𝑷⟹𝑸
CONVERSE
CONTRAPOSITVE
INVERSE

45. 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.

46. Example 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.”

47. 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.

48. “If it is raining very hard today, then classes are suspended.”
Example
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.”

49. 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.

50. Example 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.”

51. 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”

52. Summary of Truth tables𝑷 𝑸
𝑷∧𝑸
𝑷∨𝑸
𝑷⨁𝑸
𝑷⟹𝑸
𝑷⟺𝑸 ¬𝑷 ¬𝑸 T F

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

54. 𝟐 𝑵 If there are N propositions then the number of rows is 𝑷 ¬𝑷 T F 𝑷𝑸
𝑷⟹𝑸 T F

55. Tips in constructing truth tables
See the
pattern?

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

57. What's the truth table for ¬𝑷∨𝑸

58. Tautology, contradiction and contingency
A compound proposition that is
ALWAYS true – TAUTOLOGY
ALWAYS false – CONTRADICTION
SOMETIMES true – CONTINGENCY

59. Example
Consider the following propositions and determine if it is a tautology, contradiction or a contingency.
𝑝∨¬𝑝
𝑝∧¬𝑝

60. “Two propositions 𝑝,𝑞 are logically equivalent if 𝑝⟺𝑞 is a tautology.”
Show that ¬(𝑝∨𝑞) and ¬𝑝∧¬𝑞 are logically equivalent.
Construct the truth tables for p, q, 𝑝∨𝑞 ,¬ 𝑝∨𝑞 ,¬𝑝,¬𝑞 and ¬𝑝∧¬𝑞

61. SOME logical equivalences
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

62. SOME logical equivalences
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

63. On Implications
𝑝⇒𝑞
Hypothesis
Conclusion
CONVERSE:
𝑞⇒𝑝
INVERSE:
¬𝑝⇒¬𝑞
CONTRAPOSITIVE:
¬𝑞⇒¬𝑝
Show to the class that the above statement is true by constructing the truth tables of an implication and its contrapositive.
“An implication is always logically equivalent to its own contrapositive.”

64. When is mathematical reasoning correct?
Inductive
Reasoning
Deductive Reasoning
Differentiate Inductive and Deductive Reasoning.

65. Mathematical jargons
THEOREM
PROOF
AXIOM
RULES OF INFERENCE

66. Some rules of inference
Rule of Inference
Name 𝑝
_______
∴𝑝∨𝑞
ADDITION ¬𝑞
𝑝⇒𝑞
______
∴¬𝑝
MODUS TOLLENS
(the mode of denying)
𝑝∧𝑞 ∴𝑝
SIMPLIFICATION
𝑞⇒𝑟
________
∴𝑝⇒𝑟
HYPOTHETICAL SYLLOGISM 𝑞
CONJUNCTION
𝑝∨𝑞 ¬𝑝 ∴𝑞
DISJUNCTIVE SYLLOGISM
MODUS PONENS
(the mode of affirming)

67. examples
Identify the rules of inference used in each of the following arguments.
Anna is a human resource management major. Therefore, Anna is either a human resource management major or a computer applications major.
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.
1) Addition 2) Modus Ponens

68. examples
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.
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.
3) Modus tollens 4) Hypothetical Syllogism

69. Fallacy of affirming conclusion Fallacy of denying the hypothesis
Types of fallacies
Fallacy of affirming conclusion
Fallacy of denying the hypothesis
Begging the question or circular reasoning
Differentiate the three types of fallacies. Wr

70. Fallacy of affirming the conclusion!
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!

71. Fallacy of denying the hypothesis!
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!

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