Let X be the set of all real-valued functions x on the interval [0,1], and let $\displaystyle x\leqslant y mean x(t) \leqslant y(t) for all t\in [0,1] $. show that this defines partial odering. Is it a total odering? does X have a maximal elements?
Let X be the set of all real-valued functions x on the interval [0,1], and let $\displaystyle x\leqslant y mean x(t) \leqslant y(t) for all t\in [0,1] $. show that this defines partial odering. Is it a total odering? does X have a maximal elements?
Let $\displaystyle x,y,z:\left[0,1\right]\rightarrow \mathbb R$. You have to show that
a) $\displaystyle x\leqslant x$;
b) If $\displaystyle x\leqslant y$ and $\displaystyle y\leqslant x$ then $\displaystyle x(t)=y(t)\, \forall t\in:\left[0,1\right]$;
c) If $\displaystyle x\leqslant y$ and $\displaystyle y \leqslant z$ then $\displaystyle x\leqslant z$.