Truth Values (Truth Tables & Logic Statements)

Introduction In logic, truth values represent the truth or falsity of statements. In this explanation, we will learn about truth tables and logic statements to understand how truth values are determined. Truth Tables A truth table is a table used to determine the truth value of a compound statement based on the truth values of its individual components. It shows all possible combinations of truth values for the individual components and the resulting truth value of the compound statement. Logic Statements A logic statement is an assertion that can be either true or false. It consists of one or more logical operators that connect the individual components of the statement. Logical operators include "and" (conjunction), "or" (disjunction), and "not" (negation). Conjunction (AND) The conjunction operator, denoted by ∧ (caret), returns true if and only if both the individual components of the statement are true; otherwise, it returns false. The following truth table illustrates the conjunction: p q p ∧ q true true true true false false false true false false false false Disjunction (OR) The disjunction operator, denoted by ∨ (vee), returns true if at least one of the individual components of the statement is true; otherwise, it returns false. The following truth table illustrates the disjunction: p q p ∨ q true true true true false true false true true false false false Negation (NOT) The negation operator, denoted by ¬ (tilde), reverses the truth value of the individual component of the statement. If the component is true, the negation returns false, and vice versa. The following truth table illustrates the negation: p ¬p true false false true Conclusion Understanding truth tables and logic statements is fundamental in logic and problem-solving. They help us determine the truth values of compound statements and reason logically. Remember to consider all possible combinations of truth values and apply the appropriate logic operators to obtain accurate results.

Logical Operators (Truth Tables & Logic Statements)

AND Operator (&&) A B A && B true true true true false false false true false false false false OR Operator (||) A B A || B true true true true false true false true true false false false NOT Operator (!) A !A true false false true Logic Statements:

• A AND B is true if both A and B are true.
• A OR B is true if at least one of A or B is true.
• NOT A is true if A is false, and false if A is true.

Compound Statements

Truth Tables p q p AND q p OR q NOT p true true true true false true false false true false false true false true true false false false false true Logic Statements A compound statement combines multiple individual statements using logical operators such as AND, OR, and NOT. Here are some examples:

• p AND q: The statement is true if both p and q are true.
• p OR q: The statement is true if either p or q (or both) are true.
• NOT p: The statement is true if p is false, and vice versa.