one one and onto

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March 19th 2011, 08:35 AM
ayushdadhwal

one one and onto

is monotonicity is sufficient condition for proving one one .do every function must satisfy this condition.
also what is best method to prove onto
March 19th 2011, 08:49 AM
TheEmptySet

Quote:

Originally Posted by ayushdadhwal

is monotonicity is sufficient condition for proving one one .do every function must satisfy this condition.
also what is best method to prove onto

If by monotone increasing you mean (the opposite for decreasing) the important part is the strict inequality.

$x_1 < x_2 \implies f(x_1) < f(x_2)$ then this is sufficient to prove 1-1.

Suppose $f$ is monotone. Now suppose by way of contradiction that $f$ is not 1-1 then there exists $x_1 \ne x_2$ such that $f(x_1)=f(x_2)$ without loss of generality assume that $x_1 < x_2$ then by the monotone property we have that

$f(x_2) = f(x_1) < f(x_2)=f(x_1)$ but this is a contradiction!

Onto is false consider any piecewise increasing function with jump discontinuities.
March 19th 2011, 09:09 AM
ayushdadhwal

sir good evening
sir suppose i choose a function with restricted domain{1,2,3,4) such that f(1)=2 f(2)=1 f(3)=6 f(4)=-3 (ie point graph will be formed)
now this function is one one but not neither decreasing nor increasing .little bit doubt sir please clear
March 20th 2011, 11:35 PM
Drexel28

Quote:

Originally Posted by ayushdadhwal

sir good evening
sir suppose i choose a function with restricted domain{1,2,3,4) such that f(1)=2 f(2)=1 f(3)=6 f(4)=-3 (ie point graph will be formed)
now this function is one one but not neither decreasing nor increasing .little bit doubt sir please clear

I think there is a slight bit of confusion going on here. It's true, for example, that for continuous $f:\mathbb{R}\to\mathbb{R}$ one has that strict monotonicity is equivalent to injectiveness.