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Perform fuzzy arithmetic

To perform fuzzy arithmetic operations, the fuzzy operands (input fuzzy sets
`A`

and `B`

) must be *convex fuzzy
sets*. A fuzzy set is convex if, for each pair of points
*x*_{1} and
*x*_{2} in the universe of discourse
`X`

and *λ*∈[0,1].

$$\mu \left(\lambda {x}_{1}+\left(1-\lambda \right){x}_{2}\right)\ge \mathrm{min}\left(\mu \left({x}_{1}\right),\mu \left({x}_{2}\right)\right)$$

An *α-cut* of a fuzzy set is the region in the universe of discourse
for which the fuzzy set has a specific membership value, *α*. For a convex
fuzzy set, every *α*-cut defines a continuous region in the universe of
discourse.

`fuzarith`

uses the continuous regions defined by the
*α*-cuts of fuzzy sets `A`

and `B`

to
compute the corresponding *α*-cut of the output fuzzy set
`C`

. To do so, `fuzarith`

uses *interval
arithmetic*.

The following table shows how to compute the left and right boundaries of the output interval. Here:

[

*A*_{L}*A*] is the interval defined by the_{R}*α*-cut of fuzzy set*A*.[

*B*_{L}*B*] is the interval defined by the_{R}*α*-cut of fuzzy set*B*.[

*C*_{L}*C*] is the interval defined by the_{R}*α*-cut of fuzzy set*C*.

Interval Arithmetic Operator | Definition |
---|---|

Addition: C =
A+B |
$$\begin{array}{l}{C}_{L}={A}_{L}+{B}_{L}\\ {C}_{R}={A}_{R}+{B}_{R}\end{array}$$ |

Subtraction: C =
A-B |
$$\begin{array}{l}{C}_{L}={A}_{L}-{B}_{R}\\ {C}_{R}={A}_{R}-{B}_{L}\end{array}$$ |

Multiplication: C =
A*B |
$$\begin{array}{l}{C}_{L}=\mathrm{min}\left({A}_{L}\cdot {B}_{L},{A}_{L}\cdot {B}_{R},{A}_{R}\cdot {B}_{L},{A}_{R}\cdot {B}_{R}\right)\\ {C}_{R}=\mathrm{max}\left({A}_{L}\cdot {B}_{L},{A}_{L}\cdot {B}_{R},{A}_{R}\cdot {B}_{L},{A}_{R}\cdot {B}_{R}\right)\end{array}$$ |

Division: C =
A/B |
$$\begin{array}{l}{C}_{L}=\mathrm{min}\left(\frac{{A}_{L}}{{B}_{L}},\frac{{A}_{L}}{{B}_{R}},\frac{{A}_{R}}{{B}_{L}},\frac{{A}_{R}}{{B}_{R}}\right)\\ {C}_{R}=\mathrm{max}\left(\frac{{A}_{L}}{{B}_{L}},\frac{{A}_{L}}{{B}_{R}},\frac{{A}_{R}}{{B}_{L}},\frac{{A}_{R}}{{B}_{R}}\right)\end{array}$$ |