Math in the Modern World: Patterns, Language, and Propositional Logic

Fundamentals of Mathematics and Pattern Recognition

• Definition of Mathematics:

  • Mathematics is defined as the study of numbers and arithmetic operations.

  • It is a science involving logical reasoning, drawing conclusions from assumed premises, and strategic reason based on accepted rules, laws, or probabilities.

• Mathematics as the Study of Patterns:

  • Studying patterns provides an opportunity to observe, hypothesize, experiment, discover, and create.

  • Understanding regularities based on gathered data allows individuals to predict what comes next, estimate if the same pattern will occur when variables are altered, and extend the pattern.

  • Practical activities allow individuals to construct knowledge for themselves, offering all the ingredients for a meaningful, thought-provoking, and mentally and physically engaging mathematics curriculum.

• Patterns as Observers' Tools:

  • A pattern is an arrangement that helps observers anticipate what they might see or what happens next.

• Examples of Pattern Analysis Across Contexts and Developmental Levels:

  • A toddler separates blue blocks from red blocks.

  • A kindergarten student learns to count.

  • A first grader does skip counting.

  • A sixth grader creates patterns that cover a plane.

  • A biology student analyzes the sequence of DNA.

  • A stock trader analyzes the trend in the stock market.

• Four Major Types of Patterns:

  • Logic Patterns:

  • Usually the first to be observed in cognitive development.

  • Classifying objects comes prior to numeration.

  • Examples include classifying elements or sequences of symbols (e.g., A, B, D).

  • Number Patterns:

  • Sequences such as $2, 4, 6, 8, 10$ are among the first patterns encountered in school.

  • Helps predict future numerical events.

  • Example: Given the sequence $9, 18, 27, 36, 45, \dots$, the $n\text{th}$ term formula is $9n$

  • Geometric Patterns:

  • A motif or design depicting abstract shapes such as lines, polygons, and circles, typically repeating in a wallpaper-like manner.

  • Word Patterns:

  • Found in language structures, including morphological rules on pluralizing nouns or conjugating verbs for tense, as well as metrical rules of poetry.

  • Morphological example pairings: knife: knives, life: lives, wife: wives.

Mathematical Language and Symbolic Representation

• Definition of Language:

  • Language is a systematic means of communicating by the use of sounds or conventional symbols.

• Four Structural Components of Mathematical Language:

  • A vocabulary consisting of symbols and words.

  • A grammar comprising rules on the accurate use of these symbols and words.

  • A community of people who use, share, and understand the symbols.

  • A range of meanings that can be communicated with these symbols.

• Elements of Mathematical Language:

  • Contains nouns, pronouns, verbs, and complete sentence structures.

  • Features its own vocabulary, grammar, syntax, synonyms, negations, conventions, and abbreviations.

  • Uses symbols to express complex ideas in simpler, concise ways.

  • Examples of basic mathematical operational and relational symbols include: $+$, $-$, $*$, $<$, $>$, $=$

Foundations of Mathematical Logic

• Definition of Logic:

  • Logic is the study of the techniques and principles used to differentiate accurate reasoning from inaccurate reasoning.

• Aim of Studying Logic:

  • To establish and create standard criteria used to evaluate arguments and classify good arguments from bad ones.

• Concepts of Propositions versus Sentences:

  • Proposition:

  • A statement or assertion expressing a judgment or opinion (e.g., "the proposition that all men are created equal").

  • Equivalent or associated concepts include: theory, hypothesis, thesis, argument, premise, principle, theorem, concept, and idea.

  • Serves as the basic building block of an argument.

  • It is something that may be asserted or denied.

  • Propositions are the only language expressions capable of asserting whether something is true or false.

  • Sentence:

  • An idea expressed in words such as questions, commands, or exclamations.

  • Sentences express no judgment of right or wrong, or true or false.

  • Simple Proposition:

  • A declarative sentence subject to affirmation or denial, possessing a truth value of either true ($T$) or false ($F$), but not both.

Structural Elements and Classifications of Propositions

• Four Basic Structural Components of a Proposition:

  • Subject term ($S$): Designates the concept about which the assertion is made.

  • Predicate term ($P$): Designates the concept that is asserted or denied regarding the subject.

  • Copula ($C$): Words such as "are" and "are not". Expresses the present act of the idea; the verb "to be" is maintained in the present tense even when the assertion refers to the past or future. It determines whether the proposition is affirmative or negative.

  • Quantifiers ($Q$): Words such as "all", "no", and "some". Specifies the explicit quantity of the proposition.

• Classification of Propositions by Copula Type:

  • Categorical Proposition:

  • Employs a verb copula.

  • Example: "A 30-60-90 degree triangle is a right triangle."

  • Hypothetical Proposition:

  • Employs a non-verb copula or conditional connective structure.

  • Example: "If a triangle has a right angle, then it is a right triangle."

• Qualitative Classification of Propositions:

  • Affirmative Propositions:

  • Assert that the subject possesses the specified predicate quality.

  • Examples: "A quadrilateral has four sides.", "I am an Ilocana.", "Whales are mammals."

  • Negative Propositions:

  • Assert that the subject lacks the specified predicate quality.

  • Examples: "A right triangle has no obtuse angle.", "Tomato is not a fruit.", "Parallel lines never intersect."

• Quantitative Classification of Propositions:

  • Universal Proposition:

  • The subject term is taken in its full extent.

  • Examples: "All quads are polygons.", "Every integer is a real number."

  • Particular Proposition:

  • The subject term is taken only in a limited or particular extent.

  • Example: "Some people are not thinking."

  • Singular Proposition:

  • The subject term denotes a single specific person or item.

  • Example: "A prime number has only 2 factors."

• Standard Moods of Categorical Propositions (Combining Quantity and Quality):

  • $A$ (Universal Affirmative): Formulated as "All $x$ is $y$."

  • $E$ (Universal Negative): Formulated as "All $x$ is not $y$."

  • $I$ (Particular Affirmative): Formulated as "Some $x$ is $y$."

  • $O$ (Particular Negative): Formulated as "Some $x$ is not $y$."

Compound Propositions and Logical Connectives

• Syllogisms and Compound Statements:

  • Syllogism: The cornerstone of mathematical logic, dealing directly with hypothetical or compound propositions.

  • Compound Proposition: A statement formed by joining two or more simple categorical propositions using logical connectives.

• Overview of Logical Connectives and Operations:

  • Conjunction:

  • Connective word: "and"

  • Operational result: Conjunction ($p$ and $q$)

  • Symbol: $\land$

  • Disjunction:

  • Connective word: "or"

  • Operational result: Disjunction ($p$ or $q$)

  • Symbol: $\lor$

  • Implication (Conditional):

  • Connective word: "If…, then" / "implies"

  • Operational result: Implication ($p$ implies $q$)

  • Symbol: $\rightarrow$

  • Equivalence (Biconditional):

  • Connective word: "if and only if…" / "is equivalent to…"

  • Operational result: Equivalence ($p$ is equivalent to $q$)

  • Symbol: $\leftrightarrow$

  • Negation:

  • Connective word: "not"

  • Operational result: Negation (not $p$)

  • Symbol: $\sim$ or $\neg$

Truth Tables and Operations on Propositions

• Fundamental Definitions:

  • Truth Value: The status of a resulting compound proposition as either TRUE ($T$) or FALSE ($F$).

  • Truth Table: A comprehensive tabular diagram listing all possible truth values for combined logical propositions.

  • Tautology: A compound proposition that is true under all possible truth valuations of its component propositions.

  • Fallacy (Contradiction): A compound proposition that is false under all possible truth valuations of its component propositions.

  • Contingency: A compound proposition whose resulting truth values contain a mix of both true and false outcomes.

• Rules and Truth Tables for Specific Logical Operations:

  • Conjunction ($p \land q$):

  • Combines two conjuncts using the connective "and".

  • Rule: The conjunction of two statements is true if and only if both conjuncts are true.

  • Truth Table Values:

    • When $p = T$ and $q = T$, $p \land q = T$

    • When $p = T$ and $q = F$, $p \land q = F$

    • When $p = F$ and $q = T$, $p \land q = F$

    • When $p = F$ and $q = F$, $p \land q = F$

  • Disjunction ($p \lor q$):

  • Combines two disjuncts using the connective "or".

  • Rule: The disjunction of two statements is false if and only if both disjuncts are false.

  • Truth Table Values:

    • When $p = T$ and $q = T$, $p \lor q = T$

    • When $p = T$ and $q = F$, $p \lor q = T$

    • When $p = F$ and $q = T$, $p \lor q = T$

    • When $p = F$ and $q = F$, $p \lor q = F$

  • Implication / Conditional ($p \rightarrow q$):

  • Connects an antecedent (hypothesis) to a consequent (conclusion) in an "If…, then…" structure.

  • Rule: True in all cases except when the antecedent is true and the consequent is false. A true hypothesis cannot imply a false conclusion.

  • Truth Table Values:

    • When $p = T$ and $q = T$, $p \rightarrow q = T$

    • When $p = T$ and $q = F$, $p \rightarrow q = F$

    • When $p = F$ and $q = T$, $p \rightarrow q = T$

    • When $p = F$ and $q = F$, $p \rightarrow q = T$

  • Equivalence / Biconditional ($p \leftrightarrow q$):

  • Combines two propositions using the connective "if and only if" (abbreviated as iff).

  • Rule: The equivalence is true if both component propositions are true or both are false.

  • Truth Table Values:

    • When $p = T$ and $q = T$, $p \leftrightarrow q = T$

    • When $p = T$ and $q = F$, $p \leftrightarrow q = F$

    • When $p = F$ and $q = T$, $p \leftrightarrow q = F$

    • When $p = F$ and $q = F$, $p \leftrightarrow q = T$

  • Negation ($\sim p$):

  • Reverses the original truth value of a proposition.

  • Rule: If a proposition is true, its negation is false; if a proposition is false, its negation is true.

  • Truth Table Values:

    • When $p = T$, $\sim p = F$

    • When $p = F$, $\sim p = T$

• Master Truth Table Summary Across Operations:

  • For inputs $p = T$ and $q = T$: $\sim p = F$, $p \land q = T$, $p \lor q = T$, $p \rightarrow q = T$

  • For inputs $p = T$ and $q = F$: $\sim p = F$, $p \land q = F$, $p \lor q = T$, $p \rightarrow q = F$

  • For inputs $p = F$ and $q = T$: $\sim p = T$, $p \land q = F$, $p \lor q = T$, $p \rightarrow q = T$

  • For inputs $p = F$ and $q = F$: $\sim p = T$, $p \land q = F$, $p \lor q = F$, $p \rightarrow q = T$

• Step-by-Step Construction and Evaluation of $(p \lor q) \land \neg(p \land q)$:attacking steps:

  • Case 1: $p = T$, $q = T$

  • $p \lor q = T$

  • $p \land q = T$

  • $\neg(p \land q) = F$

  • Evaluated expression: $T \land F = F$

  • Case 2: $p = T$, $q = F$

  • $p \lor q = T$

  • $p \land q = F$

  • $\neg(p \land q) = T$

  • Evaluated expression: $T \land T = T$

  • Case 3: $p = F$, $q = T$

  • $p \lor q = T$

  • $p \land q = F$

  • $\neg(p \land q) = T$

  • Evaluated expression: $T \land T = T$

  • Case 4: $p = F$, $q = F$

  • $p \lor q = F$

  • $p \land q = F$

  • $\neg(p \land q) = T$

  • Evaluated expression: $F \land T = F$

  • Resulting Truth Column: $F, T, T, F$

  • Classification: Because the output contains both True and False values, $(p \lor q) \land \neg(p \land q)$ is classified as a contingency.

Related Conditionals

• Standard Statement Transformations:

  • Conditional Statement: Formed by the original hypothesis and conclusion ($p \rightarrow q$).

  • Converse: Formed by exchanging the hypothesis and conclusion of the conditional statement ($q \rightarrow p$).

  • Inverse: Formed by negating both the hypothesis and conclusion of the conditional statement ($\sim p \rightarrow \sim q$).

  • Contrapositive: Formed by negating both the hypothesis and conclusion of the converse statement ($\sim q \rightarrow \sim p$).

• Matrix of Related Conditionals Example:

  • Conditional: "If two angles have the same measure, then they are congruent."

  • Converse: "If two angles are congruent, then they have the same measure."

  • Inverse: "If two angles do not have the same measure, then they are not congruent."

  • Contrapositive: "If two angles are not congruent, then they do not have the same measure."

Practical Exercises and Problem Sets

• Exercise: Identification of Propositions (Page 19):

  • 1. All parallelograms are quadrilaterals.

  • 2. Rhombuses are squares.

  • 3. I’m from Vigan City.

  • 4. Triangle ABC is a right triangle.

  • 5. Draw two parallel lines that are cut by a transversal.

• Exercise: Categorical Mood Classification (A, E, I, or O) (Page 28):

  • 1. There are snakes in every forest.

  • 2. Children are never as free as a bird.

  • 3. Some crocodiles are found in the city.

  • 4. Not all lamb is tame.

  • 5. Some rhombuses are squares.

  • 6. All La Sallians are God-fearing.

• Exercise: Writing Statements in Symbolic Form (Page 32):

  • 1. If the sky is cloudy, then probably it will rain.

  • 2. A triangle is right iff it has a 90-degree angle.

  • 3. 2 is prime and even.

  • 4. You study harder or you will fail.

  • 5. Prices are going up but wages are not.

  • 6. He who lives in Pasig, lives in Metro Manila.

  • 7. Lines are coplanar iff they lie on the same plane.

  • 8. If $x + 3 \neq 5$, then $x \neq 2$

• Exercise: Translating Symbolic Logic into English (Page 33):

  • Given $p$: "She is beautiful." and $q$: "She is intelligent."

  • a. $p \land q$

  • b. $p \lor q$

  • c. $p \rightarrow q$

  • d. $\sim q \leftrightarrow \sim p$

  • e. $q \rightarrow p$

  • f. $\sim p \rightarrow q$

  • g. $\sim (p \land q)$

  • h. $\sim q \lor \sim p$

  • i. $\sim (\sim p \lor \sim q)$

• Exercise: Proposition Identification & Analysis (Page 34):

  • 1. Every triangle is a polygon.

  • 2. All right angles are congruent.

  • 3. $x$ is greater than or equal to $-3$

  • 4. If $x + 3 = 7$, then $x = 4$

  • 5. The sum of the interior angles of a triangle.

  • 6. Some rectangles are not parallelograms.

  • 7. Every equilateral triangle is isosceles.

  • 8. For all values of $a$ and $b$, $(a+b)(a-b) = a^2 - b^2$

  • 9. Bisect an angle.

  • 
    

  10. You’re watching!

• Exercise: Classification by Quantity and Quality (UA, UN, PA, PN) (Page 35):

  • 1. Every problem has a solution.

  • 2. All heroes are not immortals.

  • 3. No man is an island.

  • 4. Happiness is the gauge of success.

  • 5. Not every shiny object is gold.

  • 6. Some typhoons are not Pacific Ocean-origin.

  • 7. Alibis are never accepted.

  • 8. No nonreader will be left behind.

  • 9. Not every non-taxpayer is not paying.

  • 
    

  10. Not every uninformed citizen is ignorant.

• Exercise: Evaluating Truth Values of Conjunctions (Page 39):

  • 1. 3 is odd and prime.

  • 2. 5 is a factor and a multiple of 25.

  • 3. 3 and -3 is a root of $x^2 + 6x + 9 = 0$

  • 4. The quadratic function $y = x^2$ is symmetric with respect to the x-axis and has a maximum value.

• Exercise: Evaluating Truth Values of Disjunctions (Page 41):

  • 1. Tarsiers are nocturnal primates or arboreal mammals.

  • 2. A male seahorse is a fish or female seahorse carries the eggs until they release into the water.

  • 3. Platypus is a fish or it lays eggs.

  • 4. A group of dogs is a herd or a group of horses is a flock.

• Exercise: Evaluating Truth Values of Implications (Page 43):

  • 1. If $2 + 5 = 7$, then $(2)(5) = 10$

  • 2. If $2 + 5 = 7$, then $2 - 5 = 7$

  • 3. If $2 + 5 = 7$, then $7 - 5 = 2$

  • 4. If $2 + 7 = 5$, then $2 - 5 = 7$

• Exercise: Evaluating Truth Values of Equivalences (Page 45):

  • 1. A square is a polygon if and only if the square is a rectangle.

  • 2. A right triangle is a triangle iff the sum of the lengths of its sides is equal to the length of the hypotenuse.

  • 3. The diagonals of a square are parallel iff and the square is a quadrilateral.

• Exercise: Formulating Negations and Evaluating Truth (Page 48):

  • 1. Manila is the capital city of the Phil.

  • 2. Washington D.C. is not the capital of USA.

  • 3. $1 = 2$

  • 4. Jericho Valenzuela is a student of 11C.

  • 5. All policemen are honest.

• Exercise: Evaluating Truth Values and Logical Types (Tautology, Fallacy, Contingency) (Page 54):

  • 1. $\sim (p \lor \sim q)$

  • 2. $\sim (p \land \sim q)$

  • 3. $\sim (\sim p \land q)$

  • 4. $\sim (\sim p \rightarrow q)$

  • 5. $\sim (p \rightarrow q)$

  • 6. $\sim p \rightarrow \sim q$

  • 7. $p \rightarrow \sim p$

  • 8. $p \land \sim p$

  • 9. $p \lor \sim p$

  • 
    

  10. $(p \land q) \rightarrow \sim (q \rightarrow p)$

• Exercise: Formulating Related Conditional Statements (Page 57):

  • Form the "if, then" conditional, converse, inverse, and contrapositive for:

  • 1. Vertical angles are congruent.

  • 2. Angles in a linear pair are supplementary.

  • 3. If lines lie on the same line, then they are collinear.