3.1 Propositions

In this section, you learned that a proposition is a declarative statement that is either true or false, but not both. You distinguished propositions from commands and questions, which do not have truth values. You were introduced to logical variables, such as [latex]p[/latex] and [latex]q[/latex], to represent propositions abstractly, enabling logical reasoning without relying on specific statements. You studied the basic logical connectives conjunction (and), disjunction (or), and negation (not), and learned how each affects the truth value of compound propositions. You also used truth tables to systematically evaluate logical expressions and to practice translating logical statements into programming‑style conditions. These foundational ideas underpin the logical reasoning used in programming, system design, and decision‑making in IT and business contexts.

3.2 Conditional Propositions and Logical Equivalence

In this section, you explored conditional propositions of the form “if [latex]p[/latex], then [latex]q[/latex]” and learned that such statements are false only when the hypothesis is true, and the conclusion is false. You examined the concept of vacuous truth, which occurs when a conditional statement is considered true because its hypothesis is false. You also learned to interpret conditionals in terms of sufficient and necessary conditions, clarifying what guarantees an outcome and what is required for it. The section introduced biconditional statements, which assert that two propositions are true or false together. You studied logical equivalence, including equivalences such as [latex]p \rightarrow q \equiv \neg p \lor q[/latex], and applied De Morgan’s Laws to simplify logical expressions. Finally, you learned about the contrapositive and why it is logically equivalent to the original conditional, a concept that is especially important in both proof writing and program verification.

3.3 Arguments and Rules of Inference

In this section, you learned how logical statements are combined into arguments, which consist of premises and a conclusion. You studied deductive reasoning, where conclusions must logically follow from given premises, and learned that an argument is valid if the conclusion is true whenever all premises are true. You practiced analyzing arguments using truth tables and symbolic notation. The section introduced several important rules of inference, including Modus Ponens, Modus Tollens, Hypothetical Syllogism, Disjunctive Syllogism, Addition, Simplification, and Conjunction. You also examined common logical errors, particularly the fallacy of affirming the conclusion, and learned why such reasoning is invalid. These tools are essential for verifying logical correctness in mathematical proofs, security policies, business rules, and software logic.

3.4 Quantifiers

In this section, you studied predicates, which become propositions when their variables are assigned values, and identified the domain of discourse over which variables range. You learned how universal quantifiers express statements that apply to all elements in a domain, while existential quantifiers assert that at least one element satisfies a condition. You distinguished between free variables and bound variables, and practiced interpreting quantified statements in both mathematical and real‑world contexts. The section introduced nested quantifiers, showing how the order of quantifiers can significantly change the meaning of a statement. You also learned Generalized De Morgan’s Laws for quantified statements and studied rules of inference for quantifiers, such as universal instantiation and existential generalization. These concepts are fundamental for reasoning about databases, algorithms, access control systems, and system‑wide guarantees.

3.5 Proofs

In this section, you were introduced to the structure of a mathematical system, including undefined terms, definitions, axioms, theorems, lemmas, and corollaries. You learned what constitutes a proof and why formal proof is necessary to establish mathematical truth. Several proof techniques were explored, including direct proofs, proofs by contradiction, proofs by cases, proofs of equivalence, and existence proofs. You also learned how to disprove universally quantified statements using counterexamples. Throughout the section, you applied these techniques to statements involving integers, real numbers, parity, and inequalities. These proof methods form the foundation for rigorous reasoning in mathematics and support correctness arguments in algorithms, system design, and formal verification.

3.6 Mathematical induction

In this final section, you learned the principle of mathematical induction, a proof technique used to show that a statement holds for all integers greater than or equal to a given value. You studied the two required components of an induction proof: the basis step and the inductive step, and learned the role of the inductive hypothesis. You also explored strong induction, which assumes the truth of all previous cases up to a given point. The section connected induction to practical computing concepts such as loop invariants, demonstrating how induction supports reasoning about program correctness. You applied induction to sums, factorial expressions, geometric series, divisibility properties, and counting arguments such as binary strings. You also learned about the well‑ordering principle and its connection to induction. Mathematical induction is a powerful tool for proving correctness, analyzing algorithms, and modelling predictable growth and repetition in IT and business systems.