Axioms – Equality

Axioms for Real Numbers states the following about equality:

“In modern mathematics, the relation “equals” can be used between any two “mathematical objects” of the same type, such as numbers, matrices, sets, functions, etc. To say that a = b is simply to say that the symbols a and b represent the very same object. Thus equality really belongs to logic rather than to any particular branch of mathematics. Equality always has the following fundamental properties, no matter what kinds of objects it is applied to.“

The same fundamental properties referred to in the Axioms for Real Numbers apply to the axioms of equality in Wave Numbers.  In the following statements, a, b, and c can represent any mathematical objects whatsoever.

General Properties of Equality

1. (Reflexivity) a = a.
2. (Symmetry) If a = b, then b = a.
3. (Transitivity) If a = b and b = c, then a = c.
4. (Substitution) If a = b, then b may be substituted for some or all occurrences of a as a free variable in any mathematical statement without changing that statement’s truth value.

Properties of Equality of Opposite Values

In addition, for Opposite Values, we have the following properties. The first five statements say roughly that if you start with a true equation between two Opposite Values, you can “do the same thing to both sides” and still have a true equation. The last two say that if you start with two true equations, you will still have a true equation after adding them together, multiplying them together, or dividing one by the other (provided you are not dividing by zero). Finally, all of these statements can be proved using only reflexivity of equality and substitution. In these statements, a, b, c, d represent arbitrary Opposite Values.

1. If a = b, then a + c = b + c, ac = bc, and a + ^–c = b + ^–c.
2. If a = b and c is nonzero, then a/c = b/c.
3. If a = b, then ^–a = ^–b.
4. If a = b and a and b are both nonzero, then a + 1?^? = b + 1?^?.
5. If a = b, then a² = b².
6. If a = b and c = d, then a + c = b + d, ac = bd, and a + ^–c = b + ^–d.
7. If a = b and c = d, and c and d are both nonzero, then a/c = b/d.

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