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I'm interested in equation/ inequality solving ( at the elementary, intermediate level: linear, quadratic, polynomial, absolute value, log, exponent, trig equations or inequalities ). My question is :

(1) what are exactly the legal operations when performing an equation/ inequality transformation

(2) in which case precisely the transformation produces an equivalent ( open) statement , so that one can put <=> between the two statements, in which case one is allowed to put an implication sign ( ==>) but not an equivalence sign?

Mauro ALLEGRANZA
  • 99,247
  • Well, about “legal operations” on equations, if you remember that “$a=b$” means that $a$ and $b$ are not two numbers, but rather one and the same number, you really can’t go wrong. Anything you do to one side, you must do to the other, as long as you don’t try to divide both sides by zero (undefined operation) or take the square root of a negative number (undefined function). – Lubin Mar 29 '19 at 17:26
  • @Lubin. What about a < b? – William Elliot Mar 30 '19 at 02:30

1 Answers1

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Algebraic "manipulations" on equations are based of the rules for equality.

For example, from $x=3$ we derive $x+2=5$ using the substitution axiom for functions :

$t = s → f[z/t] = f[z/s]$.

In the above case, we have to use the function $f(z) := (z+2)$.

In the case of an inequality, like e.g. $x<3$ we derive $x +2 < 5$ using the arithmetical theorem :

$a < b \leftrightarrow (a + c < b + c)$,

that we can prove from Peano axioms.

The "simplifications", i.e. the manipulation leading from $x+2=3+2$ to $x+2=5$ in the first case, and from $x+2 < 3+2$ to $x+2 < 5$ are performed using the substitution axiom for formulas :

$t = s → (\phi[z/t] → \phi[z/s])$.

Mauro ALLEGRANZA
  • 99,247