Proofs

Writing a proof is perhaps the most challenging part of mathematics. However, it can be all the more rewarding once the reader becomes familiar, as the ability to logically think critically review would have been honed by the degree one feels comfortable with writing proofs.

We shall first define a proposition, a statement of a kind that can be assigned either one of truth values, truth or false, similar to yes/no question as opposed to a question formed with an interrogative word. Then, we shall introduce a truth table, a table listing all possible cases of the propositions involved. For example, when there are 3 propositions involved, then there would be possible cases and thus 8 rows for the truth table.

However, it would be inconvenient and inefficient if we need to examine all possible cases every time we prove something. Also, as the number of proposition increases, the row of truth table would increase in an exponential manner. Therefore, we developed proof techniques, a shortcut to establish truth applicable to certain situations. Then, we would be ready to delve into mathematics of abstract nature, where we shall discuss set theory, the theory of counting elements (cardinality), relations, and functions as relations. Once we complete {crossing the bridge}, we shall conduct the analysis of real numbers}.

Let us begin this section thinking about what logic is.