1.1 Propositional Logic and Truth Tables

Propositional logic is the foundation of logical reasoning in math. It deals with statements that are either true or false, using symbols to represent complex ideas. This topic introduces the building blocks of logical arguments and proofs.

Truth tables are a key tool in propositional logic. They show all possible combinations of truth values for statements, helping us understand how different logical operators work together. This systematic approach is crucial for analyzing arguments and solving logical problems.

Propositional Basics

Fundamental Components of Propositional Logic

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• Proposition defines a declarative statement that can be either true or false
• Propositions form the building blocks of logical arguments and mathematical proofs
• Simple propositions express single, complete thoughts (The sky is blue)
• Compound propositions combine multiple simple propositions using logical connectives
• Logical connectives join propositions to create more complex statements
• Common logical connectives include and, or, not, if-then, and if and only if
• Truth table provides a systematic method to determine the truth value of compound propositions
• Truth tables list all possible combinations of truth values for the component propositions
• Each row in a truth table represents a unique combination of truth values

Constructing and Interpreting Truth Tables

• Truth tables consist of columns for each proposition and logical connective
• Number of rows in a truth table equals $2^n$, where n represents the number of distinct propositions
• Fill in truth values for simple propositions first, typically using T for true and F for false
• Evaluate compound propositions by applying logical connectives to the truth values of simple propositions
• Read truth tables from left to right, following the order of operations for logical connectives
• Use truth tables to verify logical equivalences and analyze the validity of arguments

Logical Operators

Fundamental Logical Connectives

• Conjunction (AND) combines two propositions, resulting in true only when both are true
• Conjunction symbol: $\wedge$ (p $\wedge$ q reads as "p and q")
• Disjunction (OR) combines two propositions, resulting in true when at least one is true
• Disjunction symbol: $\vee$ (p $\vee$ q reads as "p or q")
• Negation (NOT) reverses the truth value of a proposition
• Negation symbol: $\neg$ ($\neg$p reads as "not p")
• Implication (IF-THEN) relates two propositions, with the truth value determined by specific rules
• Implication symbol: $\rightarrow$ (p $\rightarrow$ q reads as "if p, then q")
• Biconditional (IF AND ONLY IF) relates two propositions, true when both have the same truth value
• Biconditional symbol: $\leftrightarrow$ (p $\leftrightarrow$ q reads as "p if and only if q")

Truth Table Representations of Logical Operators

• Conjunction truth table:

  p | q | p [∧](https://www.fiveableKeyTerm:∧) q
  T | T |   T
  T | F |   F
  F | T |   F
  F | F |   F
  

• Disjunction truth table:

  p | q | p [∨](https://www.fiveableKeyTerm:∨) q
  T | T |   T
  T | F |   T
  F | T |   T
  F | F |   F
  

• Negation truth table:

  p | [¬](https://www.fiveableKeyTerm:¬)p
  T |  F
  F |  T
  

• Implication truth table:

  p | q | p [→](https://www.fiveableKeyTerm:→) q
  T | T |   T
  T | F |   F
  F | T |   T
  F | F |   T
  

• Biconditional truth table:

  p | q | p [↔](https://www.fiveableKeyTerm:↔) q
  T | T |   T
  T | F |   F
  F | T |   F
  F | F |   T
  

Truth Table Outcomes

Special Types of Compound Propositions

• Tautology represents a compound proposition that is always true, regardless of the truth values of its components
• Tautologies have a column of all T's in their truth tables (p $\vee$ $\neg$p)
• Contradiction defines a compound proposition that is always false, regardless of the truth values of its components
• Contradictions have a column of all F's in their truth tables (p $\wedge$ $\neg$p)
• Contingency describes a compound proposition that can be either true or false, depending on the truth values of its components
• Contingencies have both T's and F's in their final column of the truth table (p $\rightarrow$ q)

Analyzing and Applying Truth Table Outcomes

• Use tautologies to establish logical truths and construct valid arguments
• Identify contradictions to avoid logical inconsistencies in reasoning
• Recognize contingencies to understand the conditional nature of certain logical statements
• Apply truth table outcomes to evaluate the validity of logical arguments and proofs
• Utilize tautologies, contradictions, and contingencies in formal logic and mathematical reasoning
• Understand the role of these outcomes in developing sound logical systems and theories