Propositional Logic - Rosen 7th Edition

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.