Propositional Logic Quiz

Study Notes

A proposition is a statement that is either true or false.
Examples:
- The Moon is made of green cheese.
- Trenton is the capital of New Jersey.
- Toronto is the capital of Canada.
- 1 + 0 = 1
- 0 + 0 = 2
Examples that are not propositions:
- Sit down!
- What time is it?
- x+1=2
- x + y = z

Uses propositional variables (p, q, r, s...) to represent propositions.
T represents always true propositions, and F represents always false propositions.
Uses compound propositions (constructed from logical connectives and other propositions).
- Negation (¬): ¬p is the negation of p.
- Conjunction (∧): p ∧ q is the conjunction of p and q (both must be true).
- Disjunction (∨): p ∨ q is the disjunction of p and q (at least one must be true).
- Implication (→): p → q is the implication of p and q (if p is true then q must be true).
- Biconditional (↔): p ↔ q is the biconditional of p and q (p is true if and only if q is true).

p ∧ q is true only when both p and q are true.
If p is "I am at home," and q is "It is raining," then p ∧ q is "I am at home and it is raining."

p ∨ q is true when at least one of p or q is true.
If p is "I am at home," and q is "It is raining," then p ∨ q is "I am at home or it is raining."

In English, "or" can have two meanings:
- Inclusive Or: (p ∨ q) Both p and q can be true.
- Exclusive Or (Xor): (p ⊕ q ) Only one of p and q can be true.

p → q means "if p, then q."
- p is the hypothesis (antecedent or premise)
- q is the conclusion (or consequence).
Example: If p is "I am at home", and q is "It is raining," then p → q is "If I am at home, then it is raining."
Implication does not require any connection between the antecedent or the consequent, their truth values determine the truth value of p → q.

Implication can be viewed as an obligation or contract:
- "If I am elected, then I will lower taxes." (If the politician is elected and does not lower taxes, then the implication is false).
- "If you get 100% on the final, then you will get an A." (If the student gets 100% on the final, but does not receive an A, then the implication is false).