Set theory implications

Question

I'm pretty new to the world of fuzzy set theory, and I am trying to understand implications. So, I am wondering if someone help tell me if the following is correct.

I am trying to find the minimum of: $$ H \rightarrow \lnot G $$ and any advice or comments would be fantastic.

My attempt is:$$ H \rightarrow G = min \{H,G\} $$ $$ H \rightarrow \lnot G = min \{H,\lnot G\} $$ $$ H \rightarrow \lnot G = \lnot (min \{H,\lnot \lnot G\}) $$

So, this give

$$ H \rightarrow \lnot G = \lnot (min \{H, G\}) $$

Which should bring me to:

$$ \mu_{H \rightarrow \lnot G} (x,y) = 1 - min \{H, G\} $$

Answer 1

Let $A$ and $B$ two sets and let $m_A$ and $m_B$ their membership functions. The definition of $A\cup_{\text{fuzzy}} B$, and $\neg_{\text{fuzzy}} B$ via their mebership functions are $$m_{A\cup B}=\max(m_A,m_B)\,\,\,\text{ and } m_{\neg B}=1-m_B.$$

Also, since $A\to B\equiv\neg A \cup B$, the analogous fuzzy definition of $A\to_{\text{fuzzy}} B\equiv \neg_{\text{fuzzy}} A \cup_{\text{fuzzy}} B$ ought to be

$$m_{A\to B}=\max(1-m_A,m_B).$$

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Now, $H\to_{\text{fuzzy}} \neg_{\text{fuzzy}} G\equiv\neg_{\text{fuzzy}} H\cup_{\text{fuzzy}} \neg_{\text{fuzzy}} G$, so we have

$$m_{H\to \neg G}=\max(1-m_H,1-m_G).$$