The undefined terms are the basic building blocks mathematical system

Proofs

Greetings everyone, in this unit we will be discussing proofs. So, what are proofs? A mathematical system consists of undefined terms, definitions and axioms. So, the undefined terms are the basic building blocks of a mathematical system. These are words that are accepted as starting concepts of a mathematical system. So for example, in creating geometry we have undefined terms such as a point and line, they are both undefined the point and the line. Now, the definition on the other hand is a proposition constructive from undefined terms in previously accepted concepts in order to create a new concept. So, we use the undefined terms and we also use previously accepted concepts to create definitions.

Now, we have two other very important definitions here we have lemmas. Well, a lemma is a small theorem which is used to prove a bigger theorem. Then we have corollaries. So, a corollary is a theorem that can be proven to be a logical
consequence of another theorem. So for example, in geometry we say if the three sides of a triangle have equal length then its angles also have equal measures. So, this is an example of a corollary.

Types of Proof

Now, the first step is we assume is true and is false. And the second step is we show that ¬p is also true. The third step, then we have that p ∧ (¬p) is true. Fourth step is -- but this is impossible, since the statement p ∧ (¬p)
is always false so that is impossible for number three. There is a contradiction. And then, that lead us to the step five, so, cannot be false and therefore it is true. So, this is what we call proof by contradiction.

Another example would be to show that the contrapositive (¬q) → (¬p) is true. Since (¬q) → (¬p) is logically equivalent to p → q , then the theorem is proved.

Alright, now we need to talk about rules of inference for quantified statements. There are basically four rules here.

So, the first one is Universal instantiation. This states ∀x ∈ D P (x) d ∈ D . Therefore, P (d) .

Alright, let us take some examples here. We have p ∨ q ∨ (¬r) , this is a clause. Versus, if we have (p ∧ q) ∨ r ∨ (¬s) . This is not a clause. Hypothesis and conclusion are written as clauses. Well, there is only one rule. p ∨ q ¬p ∨ r . Therefore, q ∨ r .

Mathematical Induction