In this chapter we discuss the concepts and methods necessary for mathematical reasoning in the courses Calculus, Linear Algebra, and Statistics.
Logic is the science studying correct reasoning and reasoning schemes in everyday situations. More concretely, logic studies the relations that lead to the acceptance of a conclusion from a given set of premises in a given context. In other words, logic is an umbrella term for studying correct reasoning systems in various contexts. Thus, there are as many logics as there are fields of study.
In this chapter, we introduce classical logic, which the most universally adopted logic for correct reasoning in mathematics. This kind of reasoning is the backbone of many mathematical proofs and statements. The main goal of this chapter is to become familiar with this correct reasoning and to recognise it in everyday situations and, especially, in mathematical situations. We first present classical propositional logic and continue with classical predicate logic.
In this course, logic mostly plays an implicit role. We learn to think about true statements and correct deductions in classical logic. What is a proper mathematical proof? What are equivalent statements?
In everyday life, logic is often used subconsciously. The art of logic is that you forget you ever learned it, and absorb it into your system.
We start with the concept of proposition. We give an impression of using a proposition as a statement that can be true or false. We show how you can compose a proposition using logical operators such as 'and', 'or', and 'not'. We also show how to use a letter as a variable representing an arbitrary proposition. We continue with truth tables that reflect our intuition of the meaning of the logical operators. With truth tables, we can see how different truth values of the propositional letters lead to different truth values of more complex expressions containing those propositional letters.
Then we come to the concept of equivalence of propositions. We also discuss the omission of parentheses on the basis of priorities of the logical operators, and the propositions verum and falsum that belong to the equivalence class of the proposition that is always true and never true respectively. The theory to be dealt with up to this point is called propositional logic.
Then we arrive at predicate logic, where we discuss the so-called quantifiers #\forall# (for all) and #\exists# (there exists). Here, we introduce sets, also known as collections (the subject of a following chapter). In this context, we also discuss the concept of induction. This part is known as predicate logic.
Finally, there is a section devoted to mathematical argumentation. This section is primarily recommended for those students who deal with mathematical proofs of the theorems in our course. We treat the three main deduction rules, modus ponens, proof by contradiction, and substitution. On the one hand, it presents three main deduction steps, referred to as inference rules, that are basic for mathematical proofs. On the other hand, it phrases the inference rules in ways that prepare students for a more formal treatment of mathematical proofs. These formal treatments make their way for mathematicians to the use of proof assistance software and formal proof checkers that gradually become widely available and user friendly.
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