Week 1 - Mathematical Statements

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Discrete Mathematics

  • The study of mathematical structures that are fundamentally discrete rather than continuous.

Why discrete?

  • Computers use bits (1s and 0s) to store and process information, making discrete elements important for computing.
  • Many real-world problems can be represented using discrete mathematics.

Mathematical Statements - Definition

  • These are sentences that can be either true or false but not both.

Types of Statements

  1. Atomic (primitive): Simple, cannot be broken down further (e.g., “Three is an odd number”).

  2. Molecular (compound): Compound, composed of atomic statements connected by logical connectives (e.g., “The sun is shining, and it is cold”).

Logical Connectives

  1. “and” (∧): Conjunction (e.g., “It is raining and it is cold”).
  2. “or” (∨): Disjunction (e.g., “100 is a multiple of 5 or 10”).
  3. “if…then” (⟶): Implication/Conditional (e.g., “If a number is odd then it is prime”).
  4. “if and only if” (⟷): Equivalence/Bi-conditional (e.g., “A rectangle is a square if and only if all its four sides are equal”).
  5. “not” (¬): Negation (e.g., “Seven is not greater than fourteen”).

Propositional Variables

  • Represented by P, Q, R, S, etc., and can stand for any statement.

Truth Condition

  • P∧Q is TRUE when both P and Q are TRUE
  • P∨Q is TRUE when P or Q or both are TRUE
  • P⟶Q is TRUE when P is FALSE, or Q is TRUE or both
  • P⟷Q is TRUE when P and Q are both TRUE or both FALSE.
  • ~P is TRUE when P is FALSE

Truth Table

  • Displays the relationships between the truth values of propositions.

Implication

  • P ⟶ Q is true when P is false, or Q is true, or both.

Converse & Contrapositive

  • Converse: Q ⟶ P
  • Contrapositive: ¬Q ⟶ ¬P

If and only if

  • P ⟷ Q is logically equivalent to (P ⟶Q) ∧ (Q ⟶P)
  • P is necessary for Q means Q ⟶P
  • P is sufficient for Q means P ⟶Q
  • If P is necessary and sufficient for Q, then P ⟷ Q

Equivalence

  • P ⟷ Q is true when P and Q are both true or both false.

Predicates

  • Express the nature of a subject (e.g., P(x): x is greater than 7).

Quantifiers

  1. Universal Quantifiers (∀): “For all” or “every” (e.g., ∀x (x ≥ 0) means every number is greater than or equal to 0).
  2. Existential Quantifiers (∃): “There exists” or “there is” (e.g., ∃x (x < 0) means there is a number less than 0).

Quantifiers and Negation

  • ¬∀x P(x) is equivalent to ∃x ¬P(x)
  • ¬∃x P(x) is equivalent to ∀x ¬P(x)

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