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High 0.0 0.0 0.3 0.9 1.0 StartLayout 1st Row upper R Subscript i Baseline equals upper A Subscript i Baseline times upper B Subscript i Baseline comma that is comma mu Subscript upper R Sub Subscript i Subscript Baseline left-parenthesis normal x comma normal y right-parenthesis equals mu Subscript upper A Sub Subscript i Subscript Baseline left-parenthesis normal x right-parenthesis normal upper Lamda mu Subscript upper B Sub Subscript i Subscript Baseline left-parenthesis normal y right-parenthesis 2nd Row italic upper A s italic an example upper R 1 equals upper A 1 times upper B 1 equals italic upper L o w times italic upper L o w 3rd Row mu Subscript upper A 1 Baseline left-parenthesis normal x right-parenthesis normal upper Lamda mu Subscript upper B 1 Baseline left-parenthesis normal y right-parenthesis equals left-parenthesis 1.0 0.6 0.0 0.0 right-parenthesis normal upper Lamda left-parenthesis 1.0 1.0 0.6 0.0 0.0 right-parenthesis 4th Row equals left-parenthesis 1.0 right-parenthesis normal upper Lamda left-parenthesis 1.0 1.0 0.6 0.0 0.0 right-parenthesis 5th Row left-parenthesis 0.6 right-parenthesis normal upper Lamda left-parenthesis 1.0 1.0 0.6 0.0 0.0 right-parenthesis 6th Row left-parenthesis 0.0 right-parenthesis normal upper Lamda left-parenthesis 1.0 1.0 0.6 0.0 0.0 right-parenthesis 7th Row left-parenthesis 0.0 right-parenthesis normal upper Lamda left-parenthesis 1.0 1.0 0.6 0.0 0.0 right-parenthesis 8th Row Blank 9th Row equals left-parenthesis 1.0 1.0 0.6 0.0 0.0 right-parenthesis 10th Row left-parenthesis 0.6 0.6 0.6 0.0 0.0 right-parenthesis 11th Row left-parenthesis 0.0 0.0 0.0 0.0 0.0 right-parenthesis 12th Row left-parenthesis 0.0 0.0 0.0 0.0 0.0 right-parenthesis EndLayout

      Similarly

      (4.35)StartLayout 1st Row upper R 1 equals Start 4 By 5 Matrix 1st Row 1st Column 1.0 2nd Column 1.0 3rd Column 0.6 4th Column 0 5th Column 0 2nd Row 1st Column 0.6 2nd Column 0.6 3rd Column 0.6 4th Column 0 5th Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 EndMatrix 2nd Row upper R 2 equals Start 4 By 5 Matrix 1st Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 2nd Row 1st Column 0 2nd Column 0 3rd Column 0.3 4th Column 0.4 5th Column 0.4 3rd Row 1st Column 0 2nd Column 0 3rd Column 0.3 4th Column 0.9 5th Column 1.0 4th Row 1st Column 0 2nd Column 0 3rd Column 0.3 4th Column 0.4 5th Column 0.4 EndMatrix 3rd Row upper R 3 equals Start 4 By 5 Matrix 1st Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 2nd Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 3rd Row 1st Column 0.1 2nd Column 0.1 3rd Column 0.1 4th Column 0 5th Column 0 4th Row 1st Column 1.0 2nd Column 0 3rd Column 0.6 4th Column 0 5th Column 0 EndMatrix EndLayout right-brace upper R equals Start 4 By 5 Matrix 1st Row 1st Column 1.0 2nd Column 1.0 3rd Column 0.6 4th Column 0 5th Column 0 2nd Row 1st Column 0.6 2nd Column 0.6 3rd Column 0.6 4th Column 0.4 5th Column 0.4 3rd Row 1st Column 0.1 2nd Column 0.1 3rd Column 0.3 4th Column 0.9 5th Column 1.0 4th Row 1st Column 1.0 2nd Column 1.0 3rd Column 0.6 4th Column 0.4 5th Column 0.4 EndMatrix

      Design Example 4.3

y overTilde equals x overTilde ring upper R period StartLayout 1st Row upper B prime equals max left-parenthesis upper A prime normal upper Lamda upper R right-parenthesis 2nd Row upper A prime normal upper Lamda upper R equals left-bracket 1 comma 0.6 comma 0.3 comma 0 right-bracket normal upper Lamda Start 4 By 5 Matrix 1st Row 1st Column 1.0 2nd Column 1.0 3rd Column 0.6 4th Column 0 5th Column 0 2nd Row 1st Column 0.6 2nd Column 0.6 3rd Column 0.6 4th Column 0.4 5th Column 0.4 3rd Row 1st Column 0.1 2nd Column 0.1 3rd Column 0.3 4th Column 0.9 5th Column 1.0 4th Row 1st Column 1.0 2nd Column 1.0 3rd Column 0.6 4th Column 0.4 5th Column 0.4 EndMatrix 3rd Row equals Start 4 By 1 Matrix 1st Row 1.0 1.0 0.6 0.0 0.0 2nd Row 0.6 0.6 0.6 0.4 0.4 3rd Row 0.1 0.1 0.3 0.3 0.3 4th Row 0.0 0.0 0.0 0.0 0.0 EndMatrix EndLayout upper B prime equals max left-parenthesis upper A prime logical-and upper R right-parenthesis equals left-bracket 1.0 comma 1.0 comma 0.6 comma 0.4 comma 0.4 right-bracket

      Similarly, by using the same procedure for the input set A′ = [0, 0.2, 1, 0.2] we obtain B ′ = max (A′ ∧ R) = [0.2, 0.2, 0.3, 0.9, 1].

      Max‐min (Mamdani) inference: In the previous section, we have seen that a rule base can be represented as a fuzzy relation. The output of a rule‐based fuzzy model is then computed by the max‐min relational composition. In this section, it will be shown that the relational calculus can be bypassed. This is advantageous, as the discretization of domains and storing of the relation R can be avoided. To show this, suppose an input fuzzy value x overTilde equals upper A prime, for which the output value B′ is given by the relational composition:

      (4.36)mu Subscript upper B Sub Superscript prime Subscript Baseline left-parenthesis normal y right-parenthesis equals max left-bracket mu Subscript upper A Sub Superscript prime Subscript Baseline left-parenthesis normal x right-parenthesis logical-and mu Subscript upper R Baseline left-parenthesis normal x comma normal y right-parenthesis right-bracket period

      After substituting for μR(x, y) from Eq. (4.33), the following expression is obtained:

      (4.37)mu Subscript upper B Sub Superscript prime Subscript Baseline left-parenthesis normal y right-parenthesis equals max left-brace mu Subscript upper A Sub Superscript prime Subscript Baseline left-parenthesis normal x right-parenthesis logical-and max left-bracket mu Subscript upper A Sub Subscript i Subscript Baseline left-parenthesis normal x right-parenthesis logical-and mu Subscript upper B Sub Subscript i Subscript Baseline left-parenthesis normal y right-parenthesis right-bracket right-brace period

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