Neccesity and sufficiency .... D&K Lemma 1.3.3 ....

[Math Amateur]

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of Lemma 1.3.3 ...

Duistermaat and Kolk"s proof of Lemma 1.3.3 reads as follows:https://www.physicsforums.com/attachments/7680In the proof of Lemma 1.3.3 we read ...

"... ... The necessity is obvious. ... ... "BUT ... how are we to interpret the concepts of "necessary"and "sufficient"in the context of an "if and only if" or two-way implication statement ...

... Can someone please explain "necessary" and "sufficient" in this context?

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***EDIT***

Basically ... as I understand the terms "sufficient" and "necessary" ...

If we have S \Longrightarrow N ... ...

then

N is a necessary condition for S

and

S is a sufficient condition for N

-----------------------------------------------------------------------------------Help will be appreciated ...

Peter

 

[castor28]

Hi Peter,

"A if B" means "$B\Rightarrow A$" : B is sufficient and A is necessary.

"A only if B" means "$A\Rightarrow B$" : B is necessary and A is sufficient.

In "A if and only if B", "the condition" refers to B.

Saying that "the condition" is necessary means $A\Rightarrow B$, saying that "the condition" is sufficient means $B\Rightarrow A$.

 

[math771]

Hello Peter,

I am not familiar with the specific book you are reading, but in general, when we say something is "necessary" in a proof, it means that it is required for the statement to hold true. In other words, without the necessary condition, the statement would not be true.

On the other hand, when we say something is "sufficient", it means that it is enough to prove the statement. In other words, if we have the sufficient condition, we can conclude that the statement is true.

In the context of an "if and only if" statement, both the necessary and sufficient conditions must be met for the statement to hold true. This means that if we have the necessary condition, the statement is true, and if we have the sufficient condition, the statement is also true.

In the proof you mentioned, the authors are stating that the necessity part of the proof is obvious, meaning that the necessary condition is easy to see and understand. They may also be implying that the sufficiency part of the proof is more complex and requires further explanation.

I hope this helps clarify the concepts of "necessary" and "sufficient" in this context. Let me know if you have any other questions or need further clarification. Good luck with your studies!