Prove that following properties hold for fuzzy sets

(i) Commutativity Law

(ii) Associativity

(iii) Distributivity

(iv) Demorgan's

(i) Commutativity Law

For two fuzzy sets \(A\) and \(B\), the commutative property states that the intersection and union of fuzzy sets are commutative. That is:

1. Intersection Commutativity: 
   \[
   A \cap B = B \cap A
   \]
   This means that the membership function for the intersection of fuzzy sets \(A\) and \(B\) is the same regardless of the order in which the sets are intersected. The membership degree of the element \(x\) in the intersection of \(A\) and \(B\) is given by:
   \[
   \mu_{A \cap B}(x) = \min(\mu_A(x), \mu_B(x))
   \]
   This holds for both \(A \cap B\) and \(B \cap A\) because \(\min(\mu_A(x), \mu_B(x)) = \min(\mu_B(x), \mu_A(x))\).

2. Union Commutativity: 
   \[
   A \cup B = B \cup A
   \]
   The membership function for the union of fuzzy sets is:
   \[
   \mu_{A \cup B}(x) = \max(\mu_A(x), \mu_B(x))
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