Component-wise inequality convention for vectors

If u,v are two vectors in $mathbf{R}^n$ , the notation u le v corresponds to the component-wise inequality: that is,

u le v Longleftrightarrow u_i le v_i , ;; i=1,ldots,n.

With our convention, we can write

u := left( begin{array}{c} 2 -1 3 end{array}right) leleft( begin{array}{c} 3 0 4 end{array}right) := v.

A similar convention applies to strict inequalities. With the vectors u,v defined above, we can write the stronger statement u < v .

Not every pair of vectors can be compared in a component-wise fashion. For example, the vectors

u := left( begin{array}{c} 2 -1 3 end{array}right) , ;; v := left( begin{array}{c} 3 0 -4 end{array}right).

do not satisfy u le v , but we cannot write u > v either.

Thus, the component-wise inequality convention induces only a partial order on vectors. This is contrast with real numbers, for which the ordinary inequality induces a total order, in the sense that any pair of real numbers can be compared.