Fuzzy Intersection is Commutative

Theorem
Fuzzy intersection is commutative.

Proof
Let $\textbf A = \left({A, \mu_A}\right)$ and $\textbf B = \left({B, \mu_B}\right)$ be fuzzy sets.

Proving Domain Equality
By the definition of fuzzy intersection the domain of $\textbf A \cap \textbf B$ is:


 * $A \cap B$

Similarly the domain of $\textbf B \cap \textbf A$ is:


 * $B \cap A$

By Intersection is Commutative:


 * $A \cap B = B \cap A$

Hence their domains are equal.

Proving Form Equality
By the definition of fuzzy intersection the membership function of $\textbf A \cap \textbf B$ is of the form:


 * $\mu:A \cap B \to \left [{0 \,.\,.\, 1}\right]$

Similarly, the membership function of $\textbf B \cap \textbf A$ is of the form:


 * $\mu:B \cap A \to \left [{0 \,.\,.\, 1}\right]$

By Intersection is Commutative this is the same as:


 * $\mu:A \cap B \to \left [{0 \,.\,.\, 1}\right]$

Hence the membership functions are of the same form.

Proving Rule Equality
Hence the membership functions have the same rule.