Axioms of an equation

The axioms of an equation in Algebra are:

#Any quantity can be added to both sides #Any quantity can be subtracted from both sides #Any quantity can be multiplied to both sides #Any quantity can divide both sides except zero

Or more simply and more generally any function may be applied to both sides of an identity and the identity will still hold. Note that unless the function is injective this is an Logical conditional, not an Logical biconditional, so the solution set may get larger. The 1st, 2nd and 4th are injective, the 3rd if we do not multiply by zero. In the case of several generalized Product (mathematics), such as a dot product, there is no injectivity.

The meaning of quantity has to be understood as any algebraical quantity.

=See also=

*axioms of an inequation *Equality (mathematics)