Fuzzy Arithmetic - Fuzzy Logic: An Introductory Course for Engineering Students - page 153

corresponding to:

The number of employees that are young and with high computer skills, is
about 6,

and that gives the truth-value t )

The compression of fuzzy quantified statements is essential for its representation
and for computing its truth-value.
Let's consider the case with relative quantifiers. The most simple case is

= Q ( Z

• Among the x ∈ X , Q are such that “ F ( x ) is P ”,

for example,

• Among the company's employees almost all are young.

This statement can be compressed to

• Q are H 's, with H ( x ) = μ P ( F ( x )) ,

corresponding to

• Almost all company's employees are young.

Finally, the last compression can be done in the form:

Z is Q ,

= | H |
|

= ( )

with Z , that gives the truth value t Q Z .

X |

Example 6.4.1 (A problem of inference) Given n statements of the compressed form
“ Z i is Q i ”, 1

n with Q i absolute or relative quantifiers, which statement of the
form “ Z i is Q ” can be inferred?
For example,

≤ i ≤

There are about 10 workers in the establishment
About half of the establishment workers are women
We search Q such that: There are Q women in the stablishment

Now, it is Q 1 = about 10, Q 2 = about half, X 1 = set of workers, X 2 = set of

women ↂ X 1 . Then the syllogism can be stated by

Z 1 is Q 1
Z 2 is Q 2

| X 1 ∪ X 2 |
|

| X 2 |
|

where Z 1 =| X 1 | , Z 2 = = . Hence the conclusion is “ Z is Q ”,

X 1 | X 1 |

with Z =| X 2 | , and it rests to compute Q .