1-2_Statements_and_Connectives

Statements

• Definition of Statements

  • A statement (or proposition) is an assertion that can have a truth value of true (T) or false (F), but not both.

  • Denoted by symbols like P, Q, R, or P1, P2,...,Pn for multiple statements.

• Truth Values

  • Each statement has a truth value which can be;

    • True (T)

    • False (F)

  • It is not necessary to determine a truth value for a sentence to be classified as a statement.

Examples of Statements

• Example 1.2.2: Evaluating Statements

  • P1: "The integer 126 is prime." – False (F)

  • P2: "The 100th digit in the decimal expansion of π is 7." – Unknown truth value

  • P3: "(6)(8) 42 is a positive number." – True (T)

  • P4: "What an interesting question!" – Not a statement (exclamation)

  • P5: "Are these sets disjoint?" – Not a statement (question)

  • P6: "This statement is false." – Paradox (no truth value)

Compound Statements

• Definition of Compound Statements

  • A compound statement is composed of one or more statements combined using logical connectives.

• Logical Connectives

  • Negation: Denoted as ¬P, means "not P".

  • Conjunction: Denoted as P ∧ Q, means "P and Q".

  • Disjunction: Denoted as P ∨ Q, means "P or Q".

  • The words but, while, and although are often translated to conjunction due to similar meanings.

Examples of Logical Connectives

• Example 1.2.4: Translating Statements

  • P: "17 is prime."

  • Q: "17 is rational."

  • R: "17 is an integer."

    • 1. "17 is not prime, but is an integer." -> (¬P) ∧ R

    • 2. "17 is either prime or irrational." -> P ∨ ¬Q

    • 3. (P ∧ R) ∨ Q -> "17 is a prime integer or rational."

    • 4. (¬P) ∨ (R ∧ ¬Q) -> "17 is either not a prime or irrational integer."

Truth Tables

• Negation:

  • P | ¬P

  • T | F

  • F | T

• Conjunction (P ∧ Q) truth table:

  • P | Q | P ∧ Q

  • T | T | T

  • T | F | F

  • F | T | F

  • F | F | F

• Disjunction (P ∨ Q) truth table:

  • P | Q | P ∨ Q

  • T | T | T

  • T | F | T

  • F | T | T

  • F | F | F

Ambiguities in Everyday Language

• Mathematical interpretation of "or" is inclusive.

  • Example: "Did you have a salad or pizza for lunch?" can mean having either or both in math context.

Parenthetical Notation

• Use parentheses for clarity in expressions.

  • Example: Write (¬P) ∨ Q to clarify instead of ¬(P ∨ Q).

Truth Values and Examples

• Example 1.2.8: Evaluating Statements about Sets

  • Let A={x ∈ N | 1 ≤ x ≤ 10} and B={2, 4, 6, 9, 12, 25}

    • P: A ⊆ B (False)

    • Q: |A ∩ B| = 6 (True)

    • Evaluation:

      • (P ∧ Q) = False

      • (P ∨ Q) = True

      • (¬P) ∧ Q = True

      • (¬P) ∨ (¬Q) = True

Tautology and Contradiction

• Definitions

  • Tautology: A compound statement that is always true (e.g., P ∨ ¬P).

  • Contradiction: A compound statement that is always false (e.g., P ∧ ¬P).

• Example 1.2.13: Truth table for P ∨ ¬P:

  • P | ¬P | P ∨ ¬P

  • T | F | T

  • F | T | T

  • Concludes: P ∨ ¬P is a tautology.

• Example 1.2.14: Analyzing compound statement

  • Determine if (P ∨ (¬P) ∧ Q) ∨ (¬P ∧ ¬Q) is a tautology, contradiction, or neither.