In this post we will address what axioms and postulates are. And then we’ll talk about demonstrating results. This principle is fundamental to the development of new mathematics.

In simple terms, we can understand it as a sequence of statements, each of which is a consequence of the previous ones or something that we know to be true. How to build something as a consequence of another(s)?

Suppose that B is a consequence of previous statements. Of the previous statements, there is a statement A. This way we have:

$$A \rightarrow B$$
LR graph; A[ A - statement ] -->|It implies| B(B - consequence) B --> C{Decision} C -->|A Implies B| D[B - Result]

Transitive relations

A transitive relationship is when we have a relationship:

$$A \rightarrow B \quad \text{e} \quad B \rightarrow C \qquad \text{implicam} \qquad A \rightarrow C$$

The objective is to show that every time $A$ occurs, $C,$ occurs as long as $A \rightarrow B$ is worth as much as $B \rightarrow C.$

Suppose it is worth $A$, $A \rightarrow B$ and $B \rightarrow C$. What we want to show is that $A \rightarrow C$ is worth. From this we obtain that:

  • As $A$, $A \rightarrow B$ are valid, we have that $B$ is valid.
  • Next we have that as $B$, $B \rightarrow C$ are valid, we have that $C$ is valid.
  • This way, we have that $A \rightarrow C$ is worth.

As previously stated in other posts, if the assumption is not true, the result may be misleading.

In this world of opinions, what is true for some may not be true for others. With this, we have to be extremely precise with the assumption of true or not.

Tautologies

In mathematics, we can assume truths to be anything that is logically true, which is a tautology.

Definition: A proposition that is true in all logical possibilities is said to be a Tautology.

Definition: If this proposition is false for all logical possibilities, it is said to be a Contradiction.

In the post truth table we saw the propositions $p$ and $\sim p.$

With this, we note that if $p$ is a tautology, then $\sim p$ is a contradiction or vice versa.

Example: For every proposition $p$, the proposition $p \vee\sim p$ is a tautology and $p \wedge\sim p$ is a contradiction.

In fact, just look at the truth table:

$p$ $\sim p$ $p \vee\sim p$ $p \wedge\sim p$
V F V F
F V V F

Logical equivalence

In mathematics, equivalence is similar to “equality”, but more generally, this is not always true. If two sets of equations have the same solutions, they are considered equivalent, but they are not necessarily the same.

Twins Equivalent Triangles

When we need to use tautologies, we need to use logical equivalences. Next we will define some terms:

Definition: Two propositions are said to be logically equivalent if they have the same truth table.

Equivalently, they have the same truth value for each of the logical possibilities.

Example: The propositions $\sim (p \vee q)$, $\sim p \wedge \sim q $ are logically equivalent.

In fact, just look at the truth table:

$p$ $q$ $p \vee q$ $\sim (p \vee q)$ $\sim p$ $\sim q$ $\sim p \wedge \sim q $
V V V F F F F
V F V F F V F
F V V F V F F
F F F V V V V

Many who are reading this post will talk. What? What does this mean? What is this logical equivalence?

Suppose the first person says the propositions.

$$x-x\neq 0 \qquad \text{e} \qquad 2 x + x \neq 4x$$

Then someone else comes and says:

  • It is not true that or $$x-x= 0 \qquad \text{ou} \qquad 2 x + x = 4x$$

In this case, both people are saying the same thing, that is:

  • Both are right or
  • Both are wrong.
  • In this case, the second person’s assumption is the same as the first person’s (line 2 of the truth table).

Therefore, the propositions $p$ and $q$ are logically equivalent, so $p \leftrightarrow q$ is a tautology. Conversely, $p \leftrightarrow q$ is a tautology, so $p$ and $q$ are logically equivalent.

Logical equivalences are very useful for demonstrations, because if two propositions are logically equivalent, they are the “same” from a logical point of view.

Suppose:

  • Two propositions $p$ and $q$ and these are logically equivalent.
  • If we need to demonstrate $p$ and we have a simpler or easier way to demonstrate $q$.
  • Then we can prove $p$ by proving $q$.

Conditional, Contra positive and Reciprocal

Some terms are very important for demonstrating results. What is conditional?

Definition: The conditional is the operation performed when in the proposition we use the connective:

  • if…
  • then…

In computing, the concept of conditional is widely used.

To represent this operator we use the symbol $\rightarrow$. Like this,

  • $p \rightarrow q$ means “if p, then q”.

In this way, we obtain the following definitions:

Definition: If $p \rightarrow q$ is a conditional, then $\sim q \rightarrow \sim p$ is said contra positive conditional.

This definition is a logical equivalence. Thus, if $p \rightarrow q$ is a conditional and the objective is to demonstrate this result, simply demonstrate the positive counter as follows:

  • denies the thesis $q$
  • this implies a non-hypothesis $p$ (which we call absurdity).

Colloquially, in mathematics we call it proof by absurdity.

In the same way, we can define:

Definition:

  • If $q \rightarrow p$ is said to be a reciprocal conditional.
  • In this way, $\sim q \rightarrow \sim p$ is said to be conditional inverse.

Axioms and Postulates

In ancient Greece, mathematicians like Euclid used the concepts of axioms and postulates a lot.

Axioms and postulates are facts accepted without demonstration. More specifically, we have no way of proving results, whose facts are accepted without a demonstration. Everything starts with the axiom, as it is the starting point of reasoning.

In the case of an axiom, it is a statement that we will assume to be true. The use of axioms in general:

  • We want to define a certain subject
  • We say which axioms this satisfies
  • With this, we deduce everything we can from such axioms.
  • In other words, everything that satisfies these axioms will also satisfy their consequences.

Examples of Axioms and Postulates

As said before, ancient Greece used the concept of axioms a lot. Next, we will list some examples of axioms:

Euclid’s postulate: through a point outside a straight line passes one, and only one, straight line which is parallel to the given straight line.

  • On the line (and also outside it) there are infinite points;
  • Two distinct points determine one, and only one, straight line;
  • Three points that are not collinear determine a single plane;
  • The shortest distance between two points is the straight line that joins these two points;

This last postulate caused a lot of confusion in the Middle Ages. From this, a popular belief was created:

  • On an inclined plane when we play a ball on this inclined plane. The shortest time covered is along a straight line;

This postulate is not true today. The shortest time covered is obtained by Brachistocrona. This result was demonstrated by Johann Bernoulli, and left the condition of an axiom or postulate. Below we show a gif obtained from wikipedia.

The conclusion is that we have to evolve a little more. It is not enough to just believe in axioms or postulates. Mathematics depends a lot on results, as said before, it’s like São Tomé, seeing is believing. In the next posts we will discuss how to demonstrate a result.