Negation, Conjunction, and Disjunction in Logic

Negation

• Definition: Negation means 'not'. It is the opposite truth value of a given statement.
• Truth Value:
  • If statement P is true, negation of P is false.
  • If statement P is false, negation of P is true.
• Truth Table for Negation:
  • P: True -> ~P: False
  • P: False -> ~P: True

Conjunction

• Definition: Conjunction means 'and'.
• Truth Value:
  • True only when both statements are true.
  • If either or both statements are false, the conjunction is false.
• Truth Table for Conjunction (P and Q):
  • P: True, Q: True -> P ∧ Q: True
  • P: True, Q: False -> P ∧ Q: False
  • P: False, Q: True -> P ∧ Q: False
  • P: False, Q: False -> P ∧ Q: False

Disjunction

• Definition: Disjunction means 'or'.
• Truth Value:
  • True if either or both statements are true.
  • False only when both statements are false.
• Truth Table for Disjunction (P or Q):
  • P: True, Q: True -> P ∨ Q: True
  • P: True, Q: False -> P ∨ Q: True
  • P: False, Q: True -> P ∨ Q: True
  • P: False, Q: False -> P ∨ Q: False

Examples

1. Conjunction Example (P and Q):

   • P: 10 > 4 (True), Q: 3 < 5 (True)
   • Since both P and Q are true, P ∧ Q is True.

2. Conjunction with Negation (not P and Q):

   • P: True, negation of P is False.
   • Q is True.
   • Since P is negated to False, not P ∧ Q is False.

3. Disjunction Example (P or not Q):

   • P is True, Q is True, not Q is False.
   • Since P is True, P ∨ not Q is True.

4. Disjunction with Double Negation (not P or not Q):

   • Both not P and not Q are False.
   • Therefore, not P ∨ not Q is False.

Key Takeaways

• Negation flips the truth value.
• Conjunction requires both statements to be true.
• Disjunction requires at least one statement to be true.