9.2 Truth Tables and Logical Connectives

Truth tables are essential tools in propositional logic, helping us analyze complex logical statements. They list all possible combinations of truth values for propositions, allowing us to evaluate the overall truth of compound statements.

Logical connectives like conjunction (AND), disjunction (OR), and negation (NOT) form the building blocks of more complex logical expressions. Understanding these connectives is crucial for constructing and interpreting truth tables, as well as for logical reasoning in general.

Truth Tables and Logical Connectives

Understanding Truth Tables and Values

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• Truth tables systematically list all possible combinations of truth values for logical statements
• Truth values represent the logical status of a proposition as either true (T) or false (F)
• Truth tables consist of columns for each proposition and rows for all possible combinations
• Construct truth tables by listing all possible combinations of truth values for the given propositions
• Use truth tables to evaluate complex logical statements and determine their overall truth value

Analyzing Logical Statements

• Tautology refers to a compound statement that is always true regardless of the truth values of its components
• Identify tautologies in truth tables by checking if all rows in the final column are true
• Contradiction describes a compound statement that is always false regardless of the truth values of its components
• Recognize contradictions in truth tables when all rows in the final column are false
• Contingency applies to statements that can be either true or false depending on the truth values of their components
• Determine contingencies by examining truth tables for a mix of true and false values in the final column

Basic Logical Connectives

Conjunction and Disjunction

• Conjunction (AND) combines two propositions, resulting in a true statement only when both components are true
• Represent conjunction using the symbol $\land$ or the word "and"
• Create truth tables for conjunctions by marking true only when both input columns are true
• Disjunction (OR) combines two propositions, resulting in a true statement when at least one component is true
• Denote disjunction using the symbol $\lor$ or the word "or"
• Construct truth tables for disjunctions by marking true when either or both input columns are true
• Inclusive OR allows for both components to be true, while exclusive OR (XOR) requires exactly one true component

Negation

• Negation reverses the truth value of a proposition
• Represent negation using the symbol $\neg$ or the word "not"
• Create truth tables for negations by flipping the truth value of the input column
• Apply negation to compound statements by using De Morgan's Laws to distribute negation across conjunctions and disjunctions
• Use double negation to cancel out the effect of two consecutive negations ($\neg\neg p$ is equivalent to $p$)

Advanced Logical Connectives

Conditional Statements

• Conditional (IF-THEN) statements express a relationship where one proposition implies another
• Represent conditionals using the symbol $\rightarrow$ or the phrase "if...then"
• Construct truth tables for conditionals by marking false only when the antecedent is true and the consequent is false
• Understand the difference between necessary and sufficient conditions in conditional statements
• Recognize variations of conditional statements (converse, inverse, contrapositive)

Biconditional Statements

• Biconditional (IF AND ONLY IF) statements express logical equivalence between two propositions
• Denote biconditionals using the symbol $\leftrightarrow$ or the phrase "if and only if"
• Create truth tables for biconditionals by marking true when both components have the same truth value
• Understand biconditionals as a combination of two conditional statements (p → q) and (q → p)
• Use biconditionals to express definitions and equivalences in mathematical and logical reasoning