Sentence examples for if the ordered from inspiring English sources

So, if the ordered pair <a,b> is in a function f and so is <a,c>, then b is the same thing as c.

Note that a pair ( f, g ) is weakly increasing if and only if the ordered pairs ( f, g ) and ( g, f ) are partially weakly increasing.

It is also proved that if the ordered group Γ has the extension property and satisfies an archimedean condition then Ω=Γ×Z with the lexicographic order has the extension property.

The pair ( T 1, T 2 ) is ρ-weakly increasing if and only if the ordered pairs ( T 1, T 2 ) and ( T 2, T 1 ) are partially ρ-weakly increasing.

In the following, we check if the ordered set of criterion functions { l 0 ( x ), l 1 ( x ), l 2 ( x ) } is an ECT-system as x ∈ ( 0, 1 ) by verifying the non-vanishing property of continuous Wronskians W [ l 0 ], W [ l 0, l 1 ], W [ l 0, l 1, l 2 ].

Theorem 5 If the ordered fuzzy quasi-metric space ( X, M, ≤ M, ∗ ) is ≤ M -complete and f : X → X is a ≤ M -nondecreasing self-map such that there is x 0 ∈ X satisfying x 0 ≤ M f x 0, then f has a fixed point.

Spatial stability existed if the rank order of trap captures was kept in time.

One wonders if the order reflects the Fed's priorities.

If the order comes in online, it costs 20 cents".

But if the orders are sent to different market makers, neither might be executed.

But if the order was to kill Bin Laden, he said, that would be illegal.