Equality

There are 3 Axioms for Equality:

• Equality is Reflexive: a=a
• Equality is Symmetric: if a=b, then b=a
• Equality is Transitive: if a=b and b=a, then a=c

These ideas may also be called: Reflexivity of Equality, Symmetry of Equality and Transitivity of Equality.

If this is all you are taught, then you will probably walk away from the Algebra course thinking, “Algebra is weird.” We don’t want that, for although Algebra is weird, it also becomes interesting if you do more than just memorize the three axioms shown above.

Math is about theories and using mechanisms so you can take those theories and make more theories. If you do some good work there could be some money–do we now have your interest?

There is a problem though, in the beginning. You start with a blank slate and you don’t have any theories to use to make more theories. How do you get started, when you have nothing?

The solution is to carefully chose some theories that you can’t prove, but you, and a group of carefully chosen scholarly friends, choose to accept them. The label “axiom” gets slapped on each one to remind you that it was never proven. After choosing your axioms carefully, you can start making more theorems and using those to make more theorems (and to make some money).

As a Side Project, we can show you that if you have Reflexivity of Equality and the Definition for Substitution, you can prove Symmetry of Equality and Transitivity of Equality.