MHB 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 $B\Rightarrow A$.