max(x,y) = 1/2(x+y+|x - y|)
For two real numbers \(x\) and \(y\), prove that \(\max(x,y) = \frac{1}{2}(x+y+|x - y|)\).
For the definitions of an upper bound and the least upper bound of a set. See This.
Proof:
Suppose \(x\) and \(y\) are two real numbers. Consider the following cases
- Case \(x = y\):
- Case \(x \neq y\): Without the loss of generality suppose that \(x > y\), then
$$
\begin{align*}
\max(x, x) &= \frac{1}{2}(x + x + |x - x|) \\
&= x
\end{align*}
$$
which is true.
$$
\begin{align*}
\max{x, y} = &\frac{1}{2}(x + y + |x - y|) \\
\max{x, y} = &\frac{1}{2}(x + y + x - y) \\
\max{x, y} = &\frac{1}{2}(2x)) \\
&= x
\end{align*}
$$
Which is also true since we assumed that (x > y).
\(\blacksquare\)
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