How does one show graphically or otherwise that two fuzzy sets A and B with membership functions $1/(1+2x)$ and $1/\sqrt{1+2x}$ respectively, satisfy De Morgan's laws?
This is a question from Neural Fuzzy Systems by Chin-Teng Lin and C. S. George Lee.I was unable to find a solution to this.Could someone try to help me with this?
I guess that the univrse is $[0,\infty]$ because the membership function has to be between $0$ and $ 1$.
Since $$f_A=\frac1{1+2x}\leq f_B=\frac1{\sqrt{1+2x}},\ \text{ if } x\geq 0$$ we have that
$$f_{A\cup B}=\max(f_A,f_B)=\frac1{\sqrt{1+2x}}=f_{B}$$ and then
$$f_{\overline {A\cup B}}=1-f_B.$$
Then $f_{\overline A}=1-f_A$ and $f_{\overline B}=1-f_B$. So $f_{\overline A}\geq f_{\overline B}$ and $$f_{\overline A\cap\overline B}=\min(f_{\overline A},f_{\overline B})=f_{\overline B}=1-f_B.$$
That is, $$f_{\overline {A\cup B}}=f_{\overline A\cap\overline B}$$ according to De Morgan's law.