In-Depth Mathematics Notes

Operations on Statements

• Mathematical Statements:
  • Assigned a truth value (TRUE or FALSE), but cannot be both.
  • Represented by lowercase letters such as $p, q, r, s$.
  • Examples:
  • $p$: $1 + 1 = 2$ (True)
  • $q$: $2 + 3 = 6$ (False)
  • $r$: "All roses are red." (Objective Statement)
  • $s$: "The Philippines has more than 7,100 islands." (True)
• Non-Mathematical Statements:
  • Examples include greetings, questions, or subjective statements.
  • e.g. "Happy Birthday!" or "I am a UP student."

Statement Connectives

• Negation:
  • Symbol: $\sim p$
  • Inverts the truth value.
• Conjunction:
  • Symbol: $p \land q$
  • True only when both $p$ and $q$ are true.
• Disjunction:
  • Symbol: $p \lor q$
  • False only when both $p$ and $q$ are false.
• Conditional:
  • Symbol: $p \rightarrow q$
  • False when $p$ is true and $q$ is false.
• Biconditional:
  • Symbol: $p \leftrightarrow q$
  • True when both have the same truth value.

Truth Value of the Connectives

• Truth tables for logical connectives:
  • Conjunction:
    | $p$ | $q$ | $p \land q$ |
    |-----|-----|------------|
    | T | T | T |
    | T | F | F |
    | F | T | F |
    | F | F | F |
  • Disjunction:
    | $p$ | $q$ | $p \lor q$ |
    |-----|-----|------------|
    | T | T | T |
    | T | F | T |
    | F | T | T |
    | F | F | F |
  • Conditional:
    | $p$ | $q$ | $p \rightarrow q$ |
    |-----|-----|-----------------|
    | T | T | T |
    | T | F | F |
    | F | T | T |
    | F | F | T |
  • Biconditional:
    | $p$ | $q$ | $p \leftrightarrow q$ |
    |-----|-----|---------------------|
    | T | T | T |
    | T | F | F |
    | F | T | F |
    | F | F | T |

Negation of Statements

• To negate a statement, add "not" appropriately:
  • Example:
  • $p$: $1 + 1 = 2$
  • Negation: $\sim p$: $1 + 1 \neq 2$

Compound Statements Negation

• Negation rules for conjunction/disjunction:
  • $\sim(p \land q) \equiv \sim p \lor \sim q$
  • $\sim(p \lor q) \equiv \sim p \land \sim q$

Statements with Quantifiers

• Negation of quantified statements:
  • "All A are B" is negated to "Some A aren’t B"
  • "Some A are B" is negated to "No A are B"

Examples of Valid Argument Forms

1. Modus Ponens:
   • If $p$, then $q$.
   • $p$ is true.
   • Therefore, $q$ is true.
2. Modus Tollens:
   • If $p$, then $q$.
   • $q$ is false.
   • Therefore, $p$ is false.
3. Syllogism:
   • If $p$, then $q$; If $q$, then $r$; Therefore, if $p$, then $r$.

Common Fallacies

• Ad Hominem: Attack on character rather than argument.
• Hasty Generalization: Generalizing from insufficient evidence.
• False Cause: Correlating unrelated events.
• Strawman Fallacy: Misrepresenting an opponent's argument to refute it.

Axiomatic Systems

• Components:
  1. Definitions (Primitive Terms)
  2. Axioms (Postulates) - Accepted truths without proof
  3. Rules of Inference (Logical reasoning rules)
• Theorems: Derived conclusions from definitions and axioms.

Early Mathematics

• Contributions from various ancient civilizations (e.g., Egyptians, Greeks).
• Transition from empirical approaches to axiomatic methods.
• Significance of abstraction and rigor in mathematics development.

The Hindu-Arabic Numeral System

• Features:
  • Positional notation, inclusion of zero.
  • Historical development from ancient systems to modern use.

Cryptography Basics

• Methods of encoding/decoding messages using numerical systems.
• Cryptography principles based on modular arithmetic.

Patterns

• Symmetry Types:
  • Reflectional, Rotational, Translational.
• Frieze Patterns: Repeating patterns with symmetry characteristics.