Elementary Logic, Statements, and Methods of Mathematical Reasoning

Foundational Concepts of Logic and Propositions

Logic is formally defined as the science of thinking and reasoning correctly.
Core Terminology Unscrambling:
- "M O F N I N I T A R O" unscrambles to INFORMATION.
- "E N L W E L O G K D" unscrambles to KNOWLEDGE.
- "I D T E D V U C E" unscrambles to DEDUCTIVE.
Acronym Recall Exercise for LOGIC:
- L: ____________________
- O: ____________________
- G: ____________________
- I: ____________________
- C: ____________________
A proposition (or statement) is defined as a declarative sentence that is either true or false, without requiring additional information.
Truth Classifications of Propositions:
- Tautology: A statement that is always true under all circumstances.
- Example: $P$ : "There are $7$ days in a week."
- Contradiction: A statement that is always false under all circumstances.
- Example: "There are $6$ months with $31$ days."
- Evaluation of Sample Propositions:
- 1. "December has $28$ days."
- 2. " $4 + 5 = 1 \times 9$ "
- 3. "Tomorrow is Thursday."
Denial:
- A denial is a statement equivalent to a negation statement.
- Example 1: Given $P$ : "Today is Tuesday", the denial (negation of $P$ ) is: "Today is not Tuesday."
- Example 2: Given $P$ : "The medicine is effective", the denial (negation of $P$ ) is: "The medicine is not effective."

Logical Connectives and Compound Statements

Table of Logical Connectives and English Counterparts

Logical connectives are operators defined mathematically by truth tables, each possessing direct English equivalents:
- Conjunction: Represented by the symbol $\land$ , translating to the English word "And".
- Disjunction: Represented by the symbol $\lor$ , translating to the English word "Or".
- Negation: Represented by the symbol $\neg$ (or $\sim$ ), translating to the English word "Not".
- Conditional: Represented by the symbol $\rightarrow$ , translating to the English phrase "If…then".
- Biconditional: Represented by the symbol $\leftrightarrow$ , translating to the English phrase "If and only if".
A compound statement is defined as the result of joining two or more statements using logical connectives.
Conjunction ( $P \land Q$ ):
- Consists of two or more statements connected by the word "and".
- The symbolic representation $P \land Q$ denotes the compound statement " $P$ and $Q$ ".
- Example with propositions $P$ : "Logic is easy" and $Q$ : "Algebra is difficult":
- Conjunction of $P$ and $Q$ ( $P \land Q$ ): "Logic is easy and Algebra is difficult."
- Conjunction of $P$ and the negation of $Q$ ( $P \land \neg Q$ ): "Logic is easy and Algebra is not difficult."
Disjunction ( $P \lor Q$ ):
- Consists of two or more statements connected by the word "or".
- Example with propositions $P$ : "Juan is honest" and $Q$ : "Maria is honest":
- Disjunction ( $P \lor Q$ ): "Juan or Maria is honest."
- Disjunction of negated propositions ( $\neg P \lor \neg Q$ ): "Juan or Maria is not honest."
- Disjunction of affirmative and negated propositions ( $P \lor \neg Q$ ): "Juan is honest or Maria is not honest."

Conditional Statements and Statement Variations

Structure of a Conditional Statement ( $P \rightarrow Q$ ):
- A conditional statement consists of a hypothesis and a conclusion, conventionally written in the "If-then" structure.
- Hypothesis ( $P$ ): The "if" clause of the statement, asserting the given facts that are assumed to be true.
- Conclusion ( $Q$ ): The "then" clause of the statement, asserting what logically follows or needs to be proven.
The Five Standard Conditional Forms:
- Conditional Statement: Expressed as "If $P$ , then $Q$ " ( $P \rightarrow Q$ ).
- Example: "If logic is easy, then algebra is difficult."
- Converse Statement: Formed by reversing the hypothesis and conclusion, resulting in "If $Q$ , then $P$ " ( $Q \rightarrow P$ ).
- Example: "If algebra is difficult, then logic is easy."
- Inverse Statement: Formed by negating both the hypothesis and conclusion, resulting in "If $\neg P$ , then $\neg Q$ " ( $\neg P \rightarrow \neg Q$ ).
- Example: "If logic is not easy, then algebra is not difficult."
- Contrapositive Statement: Formed by both reversing and negating the hypothesis and conclusion, resulting in "If $\neg Q$ , then $\neg P$ " ( $\neg Q \rightarrow \neg P$ ).
- Example: "If algebra is not difficult, then logic is not easy."
- Logical Equivalence: A conditional statement ( $P \rightarrow Q$ ) and its contrapositive ( $\neg Q \rightarrow \neg P$ ) are logically equivalent; thus, one can always be used in place of the other.
- Biconditional Statement: Formed by combining a conditional statement and its converse using the connective "if and only if", denoted as $P \leftrightarrow Q$ .
- Example: "Logic is easy if and only if algebra is difficult."
Full Exemplar of Conditional Transformations:
- Given $P$ : "A student studies regularly" and $Q$ : "The student will pass the exam."
- Conditional ( $P \rightarrow Q$ ): "If a student studies regularly, then the student will pass the exam."
- Converse ( $Q \rightarrow P$ ): "If a student passes the exam, then the student studied regularly."
- Biconditional ( $P \leftrightarrow Q$ ): "A student studies regularly if and only if the student passes the exam."
- Inverse ( $\neg P \rightarrow \neg Q$ ): "If a student does not study regularly, then the student will not pass the exam."
- Contrapositive ( $\neg Q \rightarrow \neg P$ ): "If a student does not pass the exam, then the student did not study regularly."

Geometric Applications of Conditional Statements

Segment Endpoints:
- Original Statement: "A segment has only two endpoints."
- Conditional Statement: "If it is a segment, then it has only two endpoints."
- Hypothesis: "It is a segment."
- Conclusion: "It has only two endpoints."
- Converse: "If it has only two endpoints, then it is a segment."
Line Intersection:
- Original Statement: "Two lines intersect at only one point."
- Conditional Statement: "If two lines intersect, then they intersect at only one point."
- Converse Statement: "If there is only one point of intersection, then two lines intersected."
- Contrapositive: "If they do not intersect at only one point, then two lines do not intersect."
Vertical Angles:
- Original Statement: "Vertical angles are congruent."
- Conditional Statement: "If two angles are vertical, then they are congruent."
- Converse Statement: "If two angles are congruent, then they are vertical angles."
Angle Measures and Supplementary Relations:
- Conditional Statement: "If the sum of the measures of two angles is $1800$ (meaning $180^\circ$ ), then they are supplementary."
- Converse Statement: "If two angles are supplementary, then the sum of the measures of the two angles is $1800$ (meaning $180^\circ$ )."
Congruent Segments:
- Conditional Statement: "If two segments are congruent, then they have equal lengths."
- Converse Statement: "If two segments have equal lengths, then the two segments are congruent."
Isosceles Triangles:
- Original Statement: "An isosceles triangle has $2$ equal sides."
- Conditional Statement: "If a triangle is an isosceles triangle, then it has $2$ equal sides."
- Converse Statement: "If a triangle has $2$ equal sides, then it is an isosceles triangle."
- Biconditional Statement: "A triangle is an isosceles triangle if and only if it has $2$ equal sides."
- Inverse Statement: "If a triangle is not an isosceles triangle, then it does not have $2$ equal sides."
- Contrapositive Statement: "If a triangle does not have $2$ equal sides, then it is not an isosceles triangle."

The Four Kinds of Reasoning

Intuition:
- Definition: Inference achieved by common intuition, speculation, or direct subjective impression. Intuition is the capacity to obtain information without formal proof, rational deductive thought, or learned awareness, yet it is experienced as completely factual by the observer.
- Example: After the first meeting in her Statistics class, Dina says, "I think I will like Statistics."
Analogy:
- Definition: A form of reasoning in which additional similarities between two things are inferred from a known, specific similarity; reasoning grounded entirely in comparative relationship.
- Example 1: "Tree is to leaf as flower is to petal."
- Example 2: "Hammer: Nail; comb: hair."
Deductive Reasoning:
- Definition: The cognitive and logical process of drawing valid, logically necessary conclusions directly from established premises, axioms, or a given hypothesis.
- Example 1:
- Major Premise: All men are mortal.
- Minor Premise: Luke is a man.
- Conclusion: Hence, Luke is mortal.
- Example 2:
- Major Premise: All dogs like fish.
- Minor Premise: Cyber is a dog.
- Conclusion: Therefore, Cyber likes fish.
Inductive Reasoning:
- Definition: The process of gathering specific information, empirical observations, and measurements, and then synthesizing those observations to form a generalized conjecture or rule.
- Mathematical Pattern Example (Sum of Consecutive Odd Integers):
- First term: $1 = 1$
- Sum of first two terms: $1 + 3 = 4$
- Sum of first three terms: $1 + 3 + 5 = 9$
- Sum of first four terms: $1 + 3 + 5 + 7 = 16$
- Formulation of Conjecture: By observing that $1 = 1^2$ , $4 = 2^2$ , $9 = 3^2$ , and $16 = 4^2$ , the deduced rule is that the sum of the first $n$ consecutive odd numbers is equal to $n^2$ .

Euclid of Alexandria and Deductive Axiomatic Systems

Portrait of Euclid of Alexandria

Biographical and Historical Background:
- Euclid of Alexandria lived circa $300\text{ BCE}$ (born circa $325\text{ BC}$ , died circa $265\text{ BC}$ in Alexandria, Egypt).
- Known historically as the "Father of Geometry".
- Best known for his monumental mathematical treatise The Elements, a compilation that directed the development of Western mathematics for more than $2000$ years.
Structural Contributions to Reasoning:
- Euclid did not merely document geometry; he fundamentally systematized deductive reasoning.
- The Elements represents one of the earliest comprehensive works to organize mathematics using an axiomatic deductive framework:
- Axioms: Universal, self-evident truths accepted without proof.
- Postulates: Fundamental, domain-specific geometric assumptions taken as given.
- Systematic Deduction of Theorems:
- From baseline postulates and axioms, Euclid deduced geometric theorems step by step.
- Example: Beginning with the simple postulate that "a straight line can be drawn from any point to any other point," Euclid constructed rigorous, deductive proofs establishing properties of triangles, circles, and parallel lines.

Practice Exercises and Diagnostic Problem Sets

Symbolic Translation Practice:
- Given $P$ : "I like Science" and $q$ : "I like Math":
- 1. $P \land q$ translates to: "I like Science and I like Math."
- 2. $P \lor q$ translates to: "I like Science or I like Math."
- 3. $P \land \sim q$ translates to: "I like Science and I do not like Math."
- 4. $P \lor \sim q$ translates to: "I like Science or I do not like Math."
- 5. $\sim P \land \sim q$ translates to: "I do not like Science and I do not like Math."
Geometric Statement Conversion and Analysis:
- 1. "A quadrilateral has four sides."
- Conditional: "If a polygon is a quadrilateral, then it has four sides."
  - Hypothesis: A polygon is a quadrilateral.
  - Conclusion: It has four sides.
- Converse: "If a polygon has four sides, then it is a quadrilateral."
- 2. "Two points determine a line."
- Conditional: "If there are two points, then they determine a line."
  - Hypothesis: There are two points.
  - Conclusion: They determine a line.
- Converse: "If a geometric figure is a line, then it was determined by two points."
- 3. "The intersection of a line is a point."
- Conditional: "If two lines intersect, then their intersection is a point."
  - Hypothesis: Two lines intersect.
  - Conclusion: Their intersection is a point.
- Converse: "If the intersection is a point, then it is the intersection of two lines."
- 4. "Two intersecting lines lie in one plane."
- Conditional: "If two lines intersect, then they lie in the same plane."
  - Hypothesis: Two lines intersect.
  - Conclusion: They lie in the same plane.
- Converse: "If two lines lie in the same plane, then they are intersecting lines."
- 5. "The sum of the angles of a linear pair is 180."
- Conditional: "If two angles form a linear pair, then their sum is $1800$ (meaning $180^\circ$ )."
  - Hypothesis: Two angles form a linear pair.
  - Conclusion: Their sum is $180^\circ$ .
- Converse: "If the sum of the angles is $1800$ (meaning $180^\circ$ ), then they form a linear pair."
Statement Formulations and Converses:
- 1. "Good citizens obey rules and regulations."
- Conditional: "If people are good citizens, then they obey rules and regulations."
  - Hypothesis: People are good citizens.
  - Conclusion: They obey rules and regulations.
- Converse: "If people obey rules and regulations, then they are good citizens."
- 2. "Filipinos are God-fearing people."
- Conditional: "If an individual is Filipino, then that individual is God-fearing."
  - Hypothesis: An individual is Filipino.
  - Conclusion: That individual is God-fearing.
- Converse: "If an individual is God-fearing, then that individual is Filipino."
- 3. "The sum of the measure of complementary angles is 900."
- Conditional: "If two angles are complementary, then the sum of their measures is $900$ (meaning $90^\circ$ )."
  - Hypothesis: Two angles are complementary.
  - Conclusion: The sum of their measures is $90^\circ$ .
- Converse: "If the sum of the measures of two angles is $900$ (meaning $90^\circ$ ), then they are complementary angles."
- 4. "Opposite sides of a rectangle are parallel."
- Conditional: "If a quadrilateral is a rectangle, then its opposite sides are parallel."
  - Hypothesis: A quadrilateral is a rectangle.
  - Conclusion: Its opposite sides are parallel.
- Converse: "If the opposite sides of a quadrilateral are parallel, then it is a rectangle."
- 5. "A triangle is a polygon with three sides."
- Conditional: "If a polygon is a triangle, then it has three sides."
  - Hypothesis: A polygon is a triangle.
  - Conclusion: It has three sides.
- Converse: "If a polygon has three sides, then it is a triangle."
Identification of Reasoning Types (Practice Set):
- 1. "By looking at the color of the ripe mangoes, you can already tell one is sweeter than the other." -> Intuition
- 2. "The brothers, Jose, Protacio, Mario, and Bonifacio love Geometry. The fifth brother, Juan, must love Geometry also." -> Inductive Reasoning
- 3. "Birds are noisy when they are hungry. The birds in the tree are noisy. These birds must be hungry." -> Deductive Reasoning
- 4. "Hat is to head as slippers are to feet." -> Analogy
- 5. "Roel drew five different triangles. He got the sum of the measures of the three angles of each triangle. He discovered that the sum of the measure of the three angles of any triangle is 180 degrees." -> Inductive Reasoning
Identification of Reasoning Types (Quiz Set):
- 1. "I will sleep early because I have a feeling that I will have a wonderful dream." -> Intuition
- 2. "Patricia’s mother is a very good engineer. Patricia will also become a good engineer." -> Analogy / Inductive Reasoning
- 3. "In the given $1 \times 9 = 9$ , $2 \times 9 = 18$ , $3 \times 9 = 27$ , $4 \times 9 = 36$ , $5 \times 9 = 45$ : in $18$ , $1 + 8 = 9$ ; in $27$ , $2 + 7 = 9$ ; in $36$ , $3 + 6 = 9$ ; in $45$ , $4 + 5 = 9$ . Therefore, the sum of the digits of the product of a natural number and $9$ is $9$ ." -> Inductive Reasoning
- 4. "All fruits are nutritious. Apple is a fruit therefore, it is nutritious." -> Deductive Reasoning
- 5. "I am more alert this afternoon because I ate a lot of cooked vegetables this morning." -> Intuition / Inductive Reasoning