X's-being-a-necessary-condition-for-Y is both a necessary and a sufficient condition for Y's-being-a-sufficient-condition-for-X.

Asked Jul 13 '22 at 18:19

Active Jul 13 '22 at 18:19

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X's-being-a-necessary-condition-for-Y is both a necessary and a sufficient condition for Y's-being-a-sufficient-condition-for-X.

I am unable to come up with proof for the above statement. I know it's true, but I am unable to figure out how.

Following is my approach:
X being a necessary condition for Y means that Y implies X,

To prove: Y->X is both a necessary and sufficient condition for Y's being a sufficient condition for X.

Now we can write Y's being a sufficient condition for X as Y -> X.
So can we say that Y->X is a necessary and sufficient condition for Y->X

I am not sure how to proceed. Kindly help.

asked Jul 13 '22 at 18:19

Vaibhav

I think you are done since you observed that X necessary for Y, and Y sufficient for X are the exact same implication. – paw88789 Jul 13 '22 at 19:16
See the post What is the difference between necessary and sufficient conditions – Mauro ALLEGRANZA Jul 14 '22 at 06:27
Correct: the convoluted statement of the problem amounts to: $(Y \to X) \leftrightarrow (Y \to X)$ – Mauro ALLEGRANZA Jul 14 '22 at 06:29
@MauroALLEGRANZA Thanks. – Vaibhav Jul 14 '22 at 07:43
@paw88789 Can I also prove this by contradiction? How can I proceed with a proof by contradiction? Thanks for the reply. – Vaibhav Jul 14 '22 at 07:50
Cross-posted: https://math.stackexchange.com/q/4492233/14578, https://cstheory.stackexchange.com/q/51704/5038, https://stackoverflow.com/q/72970413/781723. Please do not post the same question on multiple sites. – D.W. Jul 15 '22 at 19:54

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