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Can anybody please make me understand what's the relationship among implications in proposition logic and statements such as 


1. Necessarily but not sufficient
2. Sufficient but not necessary
3. Sufficient as well as necessary
4. Neither sufficient nor necessary (
although I am skeptical about this specific statement if it exists or not. For completeness I'm telling this)

 

I have concluded the following. Correct me if I am wrong 

 

If p is sufficient but not necessary for q is the same as p --> q

If p is necessary but not sufficient for q is same as q --> p

If p is necessary as well as sufficient for q is same as p <--> q

If p is neither sufficient nor necessary is the same as ! (p <--> q)

 

in Mathematical Logic by (15 points) | 16 views
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although I am skeptical about this specific statement if it exists or not

Why not? Seems perfectly logical to me.

If p is neither sufficient nor necessary is the same as ! (p <--> q)

I think this isn’t true though, !(p<-->q) would mean p XOR q, either p or q but not both, but neither sufficient not necessary should mean there is no relation between p and q

 

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