Podcast
Questions and Answers
What is a proposition?
What is a proposition?
A declarative sentence that is either true or false but not both.
The Moon is made of green cheese.
The Moon is made of green cheese.
False
Riyadh is the capital of Saudi Arabia.
Riyadh is the capital of Saudi Arabia.
True
1 + 0 = 1.
1 + 0 = 1.
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Sit down!
Sit down!
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X + 1 = 2 is a proposition.
X + 1 = 2 is a proposition.
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What are propositional variables denoted by?
What are propositional variables denoted by?
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What is the negation of the proposition 'Michael’s PC runs Linux'?
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?
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.
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.
The disjunction of p and q is denoted by p ______ q.
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When is a conjunction true?
When is a conjunction true?
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When is a disjunction true?
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.
