Propositions and Truth Tables Quiz

Proposition (statement), Truth Table

• Definition: A proposition is a sentence that can be definitively true or false, but not both.
• Notation: Propositions are represented using letters: P, Q, R, etc.
• Examples:
  • "For all real numbers x, x² ≥ 0" (True proposition)
  • "2 x 3 = 5" (False proposition)
  • "The integer x divides 2" (Not a proposition, as it depends on the value of x)
• Truth Values:
  • True propositions are assigned the value 1 (or T).
  • False propositions are assigned the value 0 (or F).

Truth Table

• Definition: A truth table displays the truth values of a proposition for all possible combinations of truth values of its components.
• Example: Truth table for a proposition P:

Predicate

• Definition: A predicate is a logical formula that depends on one or more free variables. The truth value of a predicate is determined by assigning specific values to its variables.

Basic Logical Connectives

a) Negation

• Notation: "¬P" or "Not P"
• Definition: The negation of a proposition P is true if P is false, and false if P is true. It essentially reverses the truth value of P.
• Truth Table:

• Examples:
  • Given P: "2 × 3 = 6", its negation ¬P is "2 × 3 ≠ 6".
  • Given Q: "√2 ∈ N" (√2 is a member of the set of natural numbers), its negation ¬Q is "√2 ∉ N" (√2 is not a member of the set of natural numbers).

b) Conjunction

• Notation: "P ∧ Q" or "P and Q"
• Definition: The conjunction of propositions P and Q is true only if both P and Q are true. Otherwise, it's false.
• Truth Table:

• Examples:
  • "π is not an integer and π > 2"
  • "f(x) is decreasing and f(x) = e^x"

c) Disjunction

• Notation: "P ∨ Q" or "P or Q"
• Definition: The disjunction of propositions P and Q is false only if both P and Q are false. Otherwise, it's true.
• Truth Table:

• Examples:
  • "π ≥ 2 or e > 1"
  • "π = 3 or e < 2"