Questions and Answers
What is a proposition?
A declarative sentence that is either true or false but not both.
The Moon is made of green cheese.
False
Riyadh is the capital of Saudi Arabia.
True
1 + 0 = 1.
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Sit down!
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X + 1 = 2 is a proposition.
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What are propositional variables denoted by?
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What is the negation of the proposition 'Michael’s PC runs Linux'?
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What is the truth value of the negation of a true proposition?
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The conjunction of p and q is denoted by p ______ q.
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The disjunction of p and q is denoted by p ______ q.
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When is a conjunction true?
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When is a disjunction true?
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Study Notes
Propositions
- A proposition is a declarative sentence that is either true or false, not both.
- Truth-value indicates whether a proposition is true (T) or false (F).
Examples of Propositions
- "The Moon is made of green cheese" is false (F).
- "Riyadh is the capital of Saudi Arabia" is true (T).
- "1 + 0 = 1" is true (T).
Non-Propositions
- "Sit down!" is an imperative and not a declarative statement, so it is not a proposition.
- "x + 1 = 2" varies in truth depending on x, thus not a proposition.
- "What time is it?" is a question, not a proposition.
Propositional Variables
- Propositional variables are denoted by letters such as p, q, r, s.
- Truth values: T for true and F for false.
Compound Propositions
- Formed by combining existing propositions using logical operators (connectives).
Logical Operators
- Negation (¬ ~): Unary operation reflecting the opposite truth value.
- Conjunction (∧): True if both propositions are true.
- Disjunction (∨): True if at least one proposition is true.
- Implication (→): Denotes a conditional relationship.
- Biconditional (↔): True if both propositions share the same truth value.
Negation
- Denoted as ￢p, meaning "It is not the case that p."
- Example: Negation of "Michael’s PC runs Linux" is "Michael's PC does not run Linux."
Truth Tables
- Used to show truth-value relationships between propositions.
- Number of rows in a truth table is 2^n, where n is the number of propositions.
Truth Table for Negation
- For a proposition p:
- If p is T, then ￢p is F.
- If p is F, then ￢p is T.
Conjunction (AND) and Disjunction (OR)
- Conjunction p ∧ q: True only when both p and q are true.
- Disjunction p ∨ q: False only when both p and q are false.
Truth Table for Two Propositions
- Displays outcomes for conjunction and disjunction:
- p | q | p ∧ q | p ∨ q
- T | T | T | T
- T | F | F | T
- F | T | F | T
- F | F | F | F
Truth Table for Three Propositions
- Evaluated similarly, with 2^3 = 8 rows:
- p | q | r | p ∧ q | p ∨ r
- Truth values vary based on combinations of T and F.
Corollary
- A disjunction is true if at least one proposition is true.
- A conjunction is true only when all propositions are true.
Example Propositions
- For p = "Rebecca’s PC has more than 16 GB free space" and q = "The processor runs faster than...":
- Conjunction: p ∧ q (true if both statements hold).
- Disjunction: p ∨ q (true if at least one statement holds).
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Description
This quiz covers the fundamentals of propositions, including their truth values, types, and logical operators. You'll learn to distinguish between propositions and non-propositions, as well as how to form compound propositions. Test your understanding of logical concepts and variables with this engaging quiz.
