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Questions and Answers
What is a proposition?
A proposition is a declarative sentence that is either true or false, but not both.
Which of the following sentences are propositions? (Select all that apply)
A conjunction is true when both propositions are false.
False
What is the notation for the negation of a proposition p?
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Express the negation of the proposition 'Vandana’s smartphone has at least 32GB of memory' in simple English.
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What is the disjunction of the propositions p and q where p is 'Rebecca’s PC has more than 16 GB free hard disk space' and q is 'The processor in Rebecca’s PC runs faster than 1 GHz'?
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The conjunction of p and q is denoted by ______.
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The conditional statement p → q is only false when p is true and q is true.
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In a conditional statement p → q, what are the terms p and q referred to as?
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Study Notes
Propositional Logic
- The rules of logic define the meaning of mathematical statements and help differentiate valid from invalid arguments.
- A proposition is a declarative sentence that is either true or false, but not both.
- Examples of propositions:
- "Washington, D.C., is the capital of the United States." (True)
- "Toronto is the capital of Canada." (False)
- "1 + 1 = 2." (True)
- "2 + 2 = 3." (False)
- Non-propositions: questions or commands, e.g., "What time is it?" or "Read this carefully."
Notation
- Propositional variables (denoted by p, q, r, s, etc.) represent propositions, similar to numerical variables.
- Truth values:
- True is denoted by T,
- False is denoted by F.
- Propositional calculus or propositional logic involves the study of propositions.
- Primary (or atomic) statements cannot be broken down further into simpler statements.
Logical Connectives
- Connectives create compound statements from primary statements; for example, "and," "or," "not."
- Inexact language can lead to ambiguity, so connectives are redefined and symbolized for clarity.
Negation
- The negation of a proposition p, denoted by ¬p (or p), asserts "It is not the case that p."
- The truth value of ¬p is the opposite of p.
- Example of negation:
- Proposition: "Vandana’s smartphone has at least 32GB of memory."
- Negation: "Vandana’s smartphone does not have at least 32GB of memory."
Conjunction
- Conjunction of propositions p and q, denoted by p ∧ q, means "p and q."
- True if both p and q are true, false otherwise.
- Example:
- p: "Rebecca's PC has more than 16 GB free hard disk space."
- q: "The processor in Rebecca's PC runs faster than 1 GHz."
- Conjunction: "Rebecca's PC has more than 16 GB free hard disk space and the processor runs faster than 1 GHz."
Disjunction
- Disjunction of propositions p and q, denoted by p ∨ q, means "p or q."
- False only when both p and q are false; true otherwise.
- Example:
- Disjunction: "Rebecca's PC has at least 16 GB free hard disk space, or the processor in Rebecca's PC runs faster than 1 GHz."
Exclusive Or
- The exclusive or of p and q, denoted by p ⊕ q, is true when exactly one of p or q is true, and false otherwise.
Conditional Statements
- A conditional statement p → q means "if p, then q."
- False only when p is true and q is false; true in all other cases.
- p is the hypothesis (or antecedent), and q is the conclusion (or consequence).
- Conditional statements assert that q holds under the condition p is true.
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Description
This quiz focuses on propositional logic as outlined in Rosen's 7th edition. It covers the rules of logic that help to define mathematical statements and differentiate between valid and invalid arguments. Understand the basics of propositions and their truth values through various examples.
