OBJECTIVE:

• Definition of classical sets and fuzzy sets.

• The various operations and properties of classical and fuzzy sets.

• Functional mapping of crisp set.

• Definition of classical relation and fuzzy relation.

• Formulation of Cartesian product.

• Description on classical and fuzzy equivalence and tolerance relations.

Non interactive fuzzy sets.

Using your intuition, plot the fuzzy men function for etc age of people.

Very young (V4) : A < 12

Young (4) : 10 $\leq$ A $\leq$ 22

Middle age (M) : 20 $\leq$ A $\leq$ 42

Old (0) : 40 $\leq$ A $\leq$ 72

Very old (Vo) 70 < A

Consider two given fuzzy sets:

A = { $\frac{1}{2} + \frac {0.3}{4} + \frac {0.5}{6} + \frac {0.2}{8}$}

B = { $\frac{0.5}{2} + \frac{0.4}{4} + \frac {0.1}{6} + \frac {1}{8}$ }

Perform union, intersection difference and complement over fuzzy sets A and B.

Solution: The two given fuzzy sets are:

A = {$ \frac{1}{2} + \frac {0.3}{4} + \frac {0.5}{6} + \frac {0.2}{8}$}

B = {$ \frac{0.5}{2} + \frac{0.4}{4} + \frac{0.1}{6} + \frac{1}{8}$}

Union:

A U B = max { MA (x) , MB (X) }

= { $\frac{1}{2} + \frac{0.4}{4} + \frac{0.}{6} + \frac{1}{8}$}

Intersection:

A $\cap$ B = min { MA (x), MB (x) }

= { $\frac{0.5}{2} + \frac{0.3}{4} + \frac{0.5}{6} + \frac{0.2}{8}$}

Complement :

A = 1 – MA (x)

= { $\frac{0}{2} + \frac{0.7}{4} + \frac{0.5}{6} + \frac{0}{8}$}

B = 1 – MB (x)

= {$ \frac{0.5}{2} + \frac{0.6}{4} + \frac{0.9}{6} + \frac{0}{8}$}

Difference:

A / B = A $\cap$ B

= {$\frac{0.5}{2} + \frac{0.3}{4} + \frac{0.5}{6} + \frac{0}{8}$}

B / A = B $\cap$ A

= { $\frac{0}{2} + \frac{0.4}{4} + \frac{0.1}{6} + \frac{0.8}{8}$}