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Questions and Answers
What is a proposition?
A proposition is a sentence that is either true or false, but not both simultaneously.
The sentence 'The integer x divides 2' is a proposition.
False
What is the truth value assigned to a true proposition?
1 or T
What is the truth value assigned to a false proposition?
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What is a predicate?
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What does the negation of P (¬P) represent?
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Match the logical connectives with their definitions:
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When is a conjunction (P ∧ Q) true?
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When is a disjunction (P ∨ Q) false?
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Study Notes
Proposition (statement), Truth Table
- Definition: A proposition is a sentence that can be definitively true or false, but not both.
- Notation: Propositions are represented using letters: P, Q, R, etc.
-
Examples:
- "For all real numbers x, x² ≥ 0" (True proposition)
- "2 x 3 = 5" (False proposition)
- "The integer x divides 2" (Not a proposition, as it depends on the value of x)
-
Truth Values:
- True propositions are assigned the value 1 (or T).
- False propositions are assigned the value 0 (or F).
Truth Table
- Definition: A truth table displays the truth values of a proposition for all possible combinations of truth values of its components.
- Example: Truth table for a proposition P:
| P |
|---|
| 1 |
| 0 |
Predicate
- Definition: A predicate is a logical formula that depends on one or more free variables. The truth value of a predicate is determined by assigning specific values to its variables.
Basic Logical Connectives
a) Negation
- Notation: "¬P" or "Not P"
- Definition: The negation of a proposition P is true if P is false, and false if P is true. It essentially reverses the truth value of P.
- Truth Table:
| P | ¬P |
|---|---|
| 1 | 0 |
| 0 | 1 |
-
Examples:
- Given P: "2 × 3 = 6", its negation ¬P is "2 × 3 ≠ 6".
- Given Q: "√2 ∈ N" (√2 is a member of the set of natural numbers), its negation ¬Q is "√2 ∉ N" (√2 is not a member of the set of natural numbers).
b) Conjunction
- Notation: "P ∧ Q" or "P and Q"
- Definition: The conjunction of propositions P and Q is true only if both P and Q are true. Otherwise, it's false.
- Truth Table:
| P | Q | P ∧ Q |
|---|---|---|
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 0 |
-
Examples:
- "π is not an integer and π > 2"
- "f(x) is decreasing and f(x) = e^x"
c) Disjunction
- Notation: "P ∨ Q" or "P or Q"
- Definition: The disjunction of propositions P and Q is false only if both P and Q are false. Otherwise, it's true.
- Truth Table:
| P | Q | P ∨ Q |
|---|---|---|
| 1 | 1 | 1 |
| 1 | 0 | 1 |
| 0 | 1 | 1 |
| 0 | 0 | 0 |
-
Examples:
- "π ≥ 2 or e > 1"
- "π = 3 or e < 2"
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Description
Test your understanding of propositions, truth values, and truth tables. Learn how to differentiate between true propositions and predicates, as well as how to construct truth tables for logical statements. This quiz will help solidify your knowledge in foundational logic concepts.
