I currently started learning about fuzzy sets as well as operators and functions that can be applied to them. As such, I came by this question:
If $A, B, C$ are an element of $F$, then show that $$(A \cap ((B \cap C) \cup (A^c \cap C^c)) \cup C^c= (A \cap B \cap C) \cup C^c$$
I tried to prove it using the properties of fuzzy sets (ie. Distributive, Associative, etc), however, I can't seem to to prove either side is equal to the other. I am hoping someone could explain to me step by step how I would go about showing this equality? Thanks in advance.
$(A \cap ((B \cap C) \cup (A^c \cap C^c)) \cup C^c = (A \cap (((B \cap C) \cup A^c) \cap ((B \cap C) \cup C^c)) \cup C^c = (A \cap (((B \cap C) \cup A^c) \cap (B \cup C^c)) \cup C^c = (A \cap (((B \cap C)\cap(B \cup C^c)) \cup (A^c \cap(B \cup C^c)))\cup C^c= (A \cap ((B \cap C) \cup (A^c \cap (B \cup C^c)))\cup C^c= ((A \cap (B \cap C)) \cup (A \cap (A^c \cap(B \cup C^c))) \cup C^c= ((A \cap (B \cap C)) \cup \emptyset) \cup C^c = (A \cap B \cap C) \cup C^c $