Lecture Notes: Logical Symbols and Implications

Recap of Previous Symbols

• Propositions:
  • Identified by letters such as P, Q, R.
• Conjunction (AND):
  • Both P and Q need to be true for the statement to be true.
• Disjunction (OR):
  • P or Q is true if at least one of them is true.
  • False only if both are false.
• Exclusive OR (XOR):
  • P is true or Q is true, but not both.
• Negation (NOT):
  • Not P is true if P is false, and vice versa.
• Conditional (If-Then):
  • If P is true, then Q must be true for the statement to be true.
  • False only when P is true and Q is false.

Introduction of New Symbol

• Biconditional (If and Only If):
  • Notated as P ↔ Q.
  • True when both P and Q are either true or false.
  • Examples:
    • "x is 3 if and only if x + 1 is 4" demonstrates true equivalency in both directions.

Implications and Their Nature

• Converse Implication:
  • Q implies P is the converse of P implies Q.
  • Not always true; does not guarantee biconditionality.
  • Example:
    • Let P be "x > 2" and Q be "x² > 4".
    • P implies Q is true, but Q implies P is false.
    • Counterexample: x = -3, as x² = 9 > 4 but x < 2.

Example Analysis

• Assuming P is true:
  • x > 2 implies x² > 4 by multiplying the inequality by x.
• Assuming Q is true:
  • x² > 4 does not necessarily imply x > 2 due to negative counterexamples.

Conclusion

• Truth tables not covered in detail; to be explored in subsequent video.