{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

Before I head out into the stormy weather so my mother can "go to church"  I want to type up just a small example of the very basic kind of proof I am trying to familiarize myself with.   It may seem all-too-trivial, but it is the simplicity of it that gives me some difficulty.

Observe the following and just consider what it might be about such a procedure that I find so "defamiliarized".  (not a word, but what else can I call it?)

If I do become more familiar with this kind of familiarity, I hope I never forget this feeling of the uninitiated acolyte.

It seems too obvious to require proof, but I find the necessary mental gymnastics is taking some getting used to.  I confess that my mathematical activities have been heavy on the computational and problem solving end, but with very little formal discipline when it comes to proving even such simple statements as the following.    It is not my intention to jest.  I am serious.  I want my brain to patiently learn these "tricks".  I need references for the details, as in when to use the word axiom as opposed to when to use the word property or even principle. 

I'll get there if I can find the humility to face my ignorance without ego-inflicted shame.

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Prove:  For all real numbers a and b, if a = b, then -a = -b.

PROOF

STATEMENTS                                                      REASONS
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1. a and b are real numbers                                 1. Hypothesis

2. a + (-a) = 0                                                  2. Axiom of additive inverses
    b + (-b) = 0

3. a + (-a) = b + (-b)                                        3. Transitive axiom of equality

4. a = b                                                            4. Hypothesis

5. a + (-a) = a + (-b)                                         5. Substitution principle

6. -a is a real number                                          6. Axiom of additive inverses

7. -a + [a +(-a)] = -a + [a +(-b)]                        7. Additive property of equality

8. -a + [a + (-a)] = (-a + a) + (-b)                      8. Associative axiom of addition

9. -a + 0 = 0 + (-b)                                            9. Axiom of additive inverses

10. Therefore, -a = -b                                         10. Identity axiom for addition


Since, for most of my fairly long life, my interaction with mathematical concepts has been strongly based in "computational, mechanical problem-solving" algorithms and techniques, having to write the formal reason for each minute operation does not come second-nature to me at all.

To me, I am studying formal pure mathematics, here, in other words, Weird Mathematics.