I am attempting to prove the following statement by proving the contrapositive.
For all natural numbers n, IF n = k * l and (k less than or equal to l) THEN k less than or equal to square root of n
I have found the contrapositive to be:
For all natural numbers n, IF k greater than the square root of n THEN n does not equal k * l OR k is greater than l
I know to prove this we must first assume the first the part to be true (that
k is greater than the square root of
n) and that can lead me to say that
k squared is greater than
I am used to approaching the proof by expressing one variable in terms of another such that I can substitute the expression in for the variable in another expression. I would then solve for another variable and prove it is an integer or something like that. In this situation, how can I take the expression from one statement to sub into another if there are no equal signs involved? I do not know how to solve for an expression being not equal to another. Am I even on the right track? If I express k squared to be greater than n, how can I even move that into another equation and still keep the idea that it is
I don't understand how I can say anything about
l without it being on the left side of the if/then statement. In the current iteration of the proof, I am only assuming
k to be greater than the square root of
n. I don't see how I can assume anything more?
Any hints on what direction I can go from here would be very appreciated.