Consider two fuzzy sets.

A = { $\frac{0.2}{1} + \frac{0.3}{2} + \frac{0.4}{3} + \frac{0.5}{4}$}

B = { $\frac{0.1}{1} + \frac{0.2}{2} + \frac{0.2}{3} + \frac{0}{4}$}

Find the algebraic sum, algebraic product, bounded and bounded difference of the given fuzzy sets.

Solution:

[A] Algebraic sum:

MA + B (x) = MA (x) + r B (x) – MA (x) . MB (x)

= { $ \frac{0.3}{1} + \frac{0.5}{2} + \frac{0.6}{3} + \frac{0.5}{4}$ }

– { $\frac{0.02}{1} + \frac{0.06}{2} + \frac{0.08}{3} $ }

= { $\frac{0.28}{1} + \frac{0.44}{2} + \frac{0.52}{3} + \frac{0.5}{4}$ }

[B] Algebraic product:

MAB (x) = MA (x) MB (x)

= { $ \frac{0.02}{1} + \frac{0.06}{2} + \frac{0.08}{3} + \frac{0}{4} $ }

[C] Bounded sum:

MA (+) B (x) = min [ 1, MA (x) + MB (x)]

= min { 1, { $\frac{0.3}{1} + \frac{0.5}{2} + \frac{0.6}{3} + \frac{0.5}{4}$}}

= $\frac{0.3}{1} + \frac{0.5}{2} + \frac{0.6}{3} + \frac{0.5}{4}$

[D]Bounded Difference:

MA OB (x) = max [ 0, MA(x) – MB(x)]

= max { 0, { $\frac{0.1}{1} + \frac{0.1}{2} + \frac{0.2}{3} + \frac{0.5}{4}$}

= { $\frac{0.1}{1} + \frac{0.1}{2} + \frac{0.2}{3} + \frac{0.5}{4}$}