One of the axioms of Boolean algebra is that 1 0 1 Why is

One of the axioms of Boolean algebra is that 1 + 0 = 1. Why is it called an axiom? What\'s the difference between that and a theorem?

Solution

Axioms:

Axioms are the things that are taken as basic, unchallenged assumptions. Depending on what area of mathematics you are working within, these may change. For instance, all of the elementary results of arithmetic are usually taken as unstated axioms in higher branches of mathematics (so you don\'t prove 1+0=1 in a Boolean course, but might in a certain other courses).

Theorems:

Theorems are conclusions that can be drawn from a set of axioms by using the rules of logic. Not every such conclusion is bestowed with the title of theorem though: usually the conclusion has to be meritorious, and often the verification is non-trivial.

Axiom vs Theorem

An axiom is a statement that is considered to be true, based on logic; however, it cannot be proven or demonstrated because it is simply considered as self-evident. Basically, anything declared to be true and accepted, but does not have any proof or has some practical way of proving it, is an axiom. It is also sometimes referred to as a postulate, or an assumption.

An axiom’s basis for its truth is often disregarded. It simply is, and there is no need to deliberate any further. However, lots of axioms are still challenged by various minds, and only time will tell if they are crackpots or geniuses.

A theorem, by definition, is a statement proven based on axioms, other theorems, and some set of logical connectives. Theorems are often proven through rigorous mathematical and logical reasoning, and the process towards the proof will, of course, involve one or more axioms and other statements which are already accepted to be true.

Theorems are often expressed to be derived, and these derivations are considered to be the proof of the expression. The two components of the theorem’s proof are called the hypothesis and the conclusion. It should be noted that theorems are more often challenged than axioms, because they are subject to more interpretations, and various derivation methods.

It is not difficult to consider some theorems as axioms, since there are other statements that are intuitively assumed to be true. However, they are more appropriately considered as theorems, due to the fact that they can be derived via principles of deduction.