one one and onto

• Mar 19th 2011, 08:35 AM
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
• Mar 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.

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

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

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

Onto is false consider any piecewise increasing function with jump discontinuities.
• Mar 19th 2011, 09:09 AM
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
• Mar 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 $\displaystyle f:\mathbb{R}\to\mathbb{R}$ one has that strict monotonicity is equivalent to injectiveness.