I need help to prove the following equality in fuzzy logic. Proof that $A\cup A^c=U$ where $U$ represents a classic set, but is valid considering the T-norm drastic $T_D=${$a$ if $b=1$ , $b$ if $a=1$ or $0$ otherwise)
If $\varphi_A$ is the pertinency function of $A$. I did
In classical logical sets,$A\cup A^c = U$ where $U$ represents the universal set, but em logical fuzzy this is not true in general, but is true when you consider the Drastic T-norm. If you have two fuzzy sets $A$ and $B$ and a classical set $U$ knowing that $\varphi_A(x) \in [0,1]$ and $\varphi_B(x) \in [0,1]$ where $\varphi_A$ and $\varphi_B$ is the degree to which the x element belongs U. Now suppose we take the element $x=3 \in U$ and $\varphi_A(3)=0.4$ and $\varphi_B(3)=1$ then by the Drastic T-norm we have that $T_D(a,b)=0.4$ because $b=1$