Max Operation is Commutative
Jump to navigation
Jump to search
Theorem
The Max operation is commutative:
- $\map \max {x, y} = \map \max {y, x}$
Proof
To simplify our notation:
- Let $\map \max {x, y}$ be (temporarily) denoted $x \overline \wedge y$
There are three cases to consider:
- $(1): \quad x \le y$
- $(2): \quad y \le x$
- $(3): \quad x = y$
$(1): \quad$ Let $x \le y$.
Then:
\(\ds x \overline \wedge y\) | \(=\) | \(\ds y\) | \(\ds = y \overline \wedge x\) |
$(2): \quad$ Let $y \le x$.
Then:
\(\ds x \overline \wedge y\) | \(=\) | \(\ds x\) | \(\ds = y \overline \wedge x\) |
$(3): \quad$ Let $x = y$.
Then:
\(\ds x \overline \wedge y\) | \(=\) | \(\ds x = y\) | \(\ds = y \overline \wedge x\) |
Thus $\overline \wedge$, that is $\max$, has been shown to be commutative in all cases.
$\blacksquare$