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I want to determine the logical converse of this result. I am confused.

The complex number $s=α+iβ$ is a solution of $f(s)=0$ and $α=1$ if and only if $g(s)≠0,h(s)=u(s)$ and $d(s)=v(s)$. Here the values of the mentioned functions are not important for the purpose of this function.

I know that if $a⇔b$, then converse of $a$⇔ converse of $b$

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1 Answer 1

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The converse is $f(s)\neq 0$ or $a\neq 1$ iff $g(s)=0$ or $h(s)\neq u(s)$ or $d(s)\neq v(s)$

This is because generally, $$\neg \wedge\equiv \vee \neg$$ where $\neg$ is the negation, $\wedge$ is "and" $\vee$ is "or"

Indeed $$\neg (f(s)=0\wedge a=1)\equiv (\neg f(s) =0)\vee (\neg a=1)$$ and $$\neg (g(s)=0\wedge h(s)=u(s)\wedge d(s)=v(s))\equiv (\neg g(s) =0)\vee (\neg h(s)=u(s))\vee (\neg d(s)=v(s))$$

It might be useful to know that $$\neg \forall\equiv \exists\neg$$ as well

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Yes, but the case: $f(s)=0$ and $a≠1$ should be included. –  ZE1 Dec 24 '12 at 10:37
    
@user53124 $A\vee B$ means "$A$" or "$B$" or "$A$ and $B$". That case is inculded in the "or" –  Nameless Dec 24 '12 at 10:37
    
Yes, and thank you. Things are now clear. –  ZE1 Dec 24 '12 at 10:39

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