I had this homework problem:
Use the following definition of absolute value to prove the given
statements: If $x$ is a real number, then the absolute of $x$, $|x|$,
is defined by: $$|x|= \left\{ \begin{array}{} x & x \geq 0 \\ -x & x < 0 \\ \end{array} \right.$$
a) For any real number $x$, $|x| \geq 0$. Moreover, $x = 0$ implies $x = 0$.
b) For any two real numbers $x$ and $y$, $|x||y| = |xy|$.
c) For any two real numbers $x$ and $y$, $|x+y| \leq |x|+|y|$.
I spent almost 2 hours on this problem unable to make any headway because I thought it wanted me to prove the first definition given the 3 statements a, b, c. Only to realize that the intended interpretation of the problem (prove a, b, c given the definition) is absolutely trivial…
It still seems possible to prove the definition from a, b, c, but I can't justify spending anymore time on this. Anyone willing to give it a shot?
Best Answer
No, there are other functions satisfying $(a)\land (b)\land (c)$. Namely, $\lvert x\rvert^{1/2}$.