rewriting implication, making the quantifiers explicit

[kluda06]

Hello! I have a question and need help

I have to re write the implication, making he quantifiers explicit.

If f is an increasing function , then f is differentiable.

so I wrote

Every increasing function is differentiable.

(hopefully i am right)

Now the next question is the statement is true or false and to prove my answer.
This is where I get confused. I am not understanding how she wants me to prove the answer. Do I have to make a problem or... we didnt get to go over this much in class. Help me out please

 

[Ishuda]

Hello! I have a question and need help

I have to re write the implication, making he quantifiers explicit.

If f is an increasing function , then f is differentiable.

so I wrote

Every increasing function is differentiable.

(hopefully i am right)

I'm not sure this is correct. You may want to read http://www.math.uri.edu/~eaton/Day5.pdf

Now the next question is the statement is true or false and to prove my answer.
This is where I get confused. I am not understanding how she wants me to prove the answer. Do I have to make a problem or... we didnt get to go over this much in class. Help me out please

I believe this question is "Is the statement 'If f is an increasing function, then f is differentiable.' true or false? Prove your answer". As a hint, note that to prove a statement false, you just need one counter example. For example the statement 'all flowers are green' is false because Texas bluebonnets are blue so there exists a flower which is not green.

 

[stapel]

I have to re write the implication, making the quantifiers explicit.

To be clear, "quantifiers" are the logical "for all (such and such)" and "there exists a (such and such)", indicated symbolically as \(\displaystyle \, \forall\, \) and \(\displaystyle \, \exists\).

If f is an increasing function, then f is differentiable.

so I wrote

Every increasing function is differentiable.

The "if" part of the "if-then" is now missing. Perhaps a better way to state this would be:

. . .For every function f, if f is increasing, then f is differentiable.

Now the next question is the statement is true or false and to prove my answer. This is where I get confused. I am not understanding how she wants me to prove the answer.

Use the logical rules, etc, like she'd shown in class and like they show in the worked examples in your book. If you think that this statement is untrue, then find a counter-example. One place (that is, one function) for which the "for all" is false makes the whole statement false. On the other hand, if you think the statement is true, then you'll need to work with the generic "Let f be an increasing function", and work through the logic and the definitions to arrive at "then f is differentiable".

Either way, your book's precise definition of "increasing" will almost certainly be crucial.