Prove an equation is always false

Question

How can I prove an equation is always false?

For example:

$b = b + 1$

is false for all values of $b$. Very simple to see.

Now given a more complicated equation, such as:

$b = \sin(\sin(b) - 0.56)$

How can I prove that some value of $b$ does (or does not) satisfy the equation? For all equations not just this one?

AKA: I am not trying to find a value of $b$, just if any $b$ can satisfy the equation.

Try to plot the right hand side of your equation, and the left hand side (in this case the left hand side is the bisector). If they intersect, there is a solution. (In this case is probably not feasible to do it by hand, but in many other cases it's helpful ) — Ant, Feb 27 '15 at 22:03
For the example you gave, there is a solution with $b$ between $-1$ and $1$. When $b=-1$, $b\le\sin(\sin(b-.56))$, and when $b=1$, $b\ge\sin(\sin(b-.56))$. Both $b$ and $\sin(\sin(b-.56))$ are continuous for $b$-values between $-1$ and $1$, so their graphs must cross. — Steve Kass, Feb 27 '15 at 22:07
You want some $b$ that makes any equation it is used in false? — mvw, Feb 27 '15 at 22:09
@mvw no, I am asking this: for a given equation (that is always false), prove that it is false for all values of b. — KobeSystem, Feb 27 '15 at 22:10

score 1 · Answer 1 · answered Feb 27 '15 at 22:08

Well it's not that simple to prove that whether there are or there are not solutions for each equation. Just think of the Fermat's Last Theorem, it took around 300 years for it to be solved. Anyway in this situation I want to use the method of contraposition (contradiction). Let's say that there is a solution b to the equation, then find some other true identity (like b must be within some range, b must be a square or so) that will contradict the initial assumption that there exist such a solution.

BTW, you won't be able to do it for you equation, since there is such a solution. $b \approx -0.98$ will be one solution according to WolframAlpha

Peter Franek · Answer 2 · 2015-02-27T22:46:17.083

In general, the question is complicated. While most mathematicians believe that human mind can decide the satisfiability of any such equations (whenever it can be decided in ZFC), there are known results that it can not be done algorithmically. Two examples:

Satisfiability of systems of quadratic equations over integers is algorithmically undecidable (this is a simple reduction from Matiyasevich's undecidability theorem)
Satisfiability of one equation in one real variable is (algorithmically) undecidable, if the equations can contain arbitrary combinations of polynomials, the sin function and $\pi$. (Wang: The Undecidability of the Existence of Zeros of Real Elementary Functions)

Prove an equation is always false

2 Answers2