Foundations of Propositional Logic and Equivalences

which systematically introduces the fundamentals of formal logic. It begins by defining logic as the study of reasoning and a proposition as a declarative sentence that is exclusively true or false. The text then outlines compound propositions formed by combining existing propositions using various logical connectives, such as negation ($\neg$), conjunction ($\land$), and disjunction ($\lor$). Furthermore, the document explains complex forms like conditional statements ($p \rightarrow q$) and bi-implications ($p \leftrightarrow q$), noting their converse, inverse, and contrapositive variations. The material also covers the use of truth tables to determine the output of compound propositions, discusses the precedence of logical operators, and defines concepts like tautology and contradiction, along with practical applications in bit operations and logic circuits.