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Generalization of some results of Elie Sanchez

Siegfried Gottwald

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Siegfried Gottwald. Generalization of some results of Elie Sanchez. Bulletin pour les sous-ensembles flous et leurs applications, 1983, 16, pp.54–60. hal-04583324

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# Generalization of some results of Elie Sanchez

Siegfried Gottwald

► To cite this version:

Siegfried Gottwald. Generalization of some results of Elie Sanchez. Bulletin pour les sous-ensembles flous et leurs applications, 1983, 16, pp.54–60. hal-04583324

HAL Id: hal-04583324

https://hal.science/hal-04583324v1

Submitted on 22 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
# GENERALIZATION OF SOME RESULTS

OF ELIE SANCHEZ

Siegfried Gottwald

Logic Group, Dept. of Philosophy,
Karl Marx University,
Leipzig, G. D. R.

**Abstract** Recently E. Sanchez has given a necessary and sufficient condition for the existence of a solution of a fuzzy equation $A*X = C$, with * the extension of a given binary operation in the universe of discourse.

Using many valued logic we generalize this result
and get a characterization also of the cases if $A*X = C$
has only nearly a solution.

1. Introduction

Just recently E. SANCHEZ /5/ was considering the prob-
lem of solving set equations A*X = B for fuzzy sets A,B
where the operation * extends an operation in the common
universe of discourse of A,X. The problem is relevant e.g.
for the arithmetics of fuzzy numbers.
GENERALIZATION OF SOME RESULTS

OF ELIE SANCHEZ

Siegfried Gottwald

Logic Group, Dept. of Philosophy,
Karl Marx University,
Leipzig, G. D. R.

**Abstract** Recently E. Sanchez has given a necessary and sufficient condition for the existence of a solution of a fuzzy equation $A*X = C$, with * the extension of a given binary operation in the universe of discourse.

Using many valued logic we generalize this result
and get a characterization also of the cases if $A*X = C$
has only nearly a solution.

# 1. Introduction

Just recently E. SANCHEZ /5/ was considering the prob-
lem of solving set equations A*X = B for fuzzy sets A,B
where the operation * extends an operation in the common
universe of discourse of A,X. The problem is relevant e.g.
for the arithmetics of fuzzy numbers.
Using as in earlier papers /2/, /3/ essentially the language and results of many valued logics we will sketch here how these means are helpful to give his results a greater generality and power.

The present results have been found during a summer school in Primorsko/Bulgaria. The author kindly acknowledges his invitation and the encouraging and fruitful conditions of this meeting, realized by the organizers.

## 2. Logical Notation

Mainly the same notation as in /3/ is used. For many valued logic, especially, we do not distinguish the connectives and their corresponding truth functions. The set of generalized truth values is the real interval [0,1]; the connectives we use are conjunctions described by

$$s \Lambda_1 t = \min(s,t),$$

$$s \Lambda_2 t = \max(0,s+t-1),$$

and implications described by

$$s \rightarrow_1 t = \min(1,1-s+t),$$

$$s \rightarrow_2 t = \begin{cases} 1, & \text{if } s \le t \\ t, & \text{otherwise.} \end{cases}$$

Of course, this implication $\rightarrow_2$ is nothing else than the $\alpha$-operation of E. SANCHEZ /4/, /5/. But, $\rightarrow_2$ is also that many valued implication operator which was used by K. GÖDEL /1/ for proving that intuitionistic propositional logic does not have a finite adequate matrix. Therefore we prefer to denote this operation as an implication.

We write $\models H$ to indicate, that formula H of our
Using as in earlier papers /2/, /3/ essentially the language and results of many valued logics we will sketch here how these means are helpful to give his results a greater generality and power.

The present results have been found during a summer school in Primorsko/Bulgaria. The author kindly acknowledges his invitation and the encouraging and fruitful conditions of this meeting, realized by the organizers.

## 2. Logical Notation

Mainly the same notation as in /3/ is used. For many valued logic, especially, we do not distinguish the connectives and their corresponding truth functions. The set of generalized truth values is the real interval [0,1]; the connectives we use are conjunctions described by

$$s \Lambda_1 t = \min(s,t) ,$$

$$s \Lambda_2 t = \max(0,s+t-1) ,$$

and implications described by

$$s \rightarrow_1 t = \min(1,1-s+t) ,$$

$$s \rightarrow_2 t = \begin{cases} 1, & \text{if } s \le t \\ t, & \text{otherwise.} \end{cases}$$

Of course, this implication $\rightarrow_2$ is nothing else than the $\alpha$-operation of E. SANCHEZ /4/, /5/. But, $\rightarrow_2$ is also that many valued implication operator which was used by K. GÖDEL /1/ for proving that intuitionistic propositional logic does not have a finite adequate matrix. Therefore we prefer to denote this operation as an implication.

We write $\models H$ to indicate, that formula H of our
many valued language always has truth value 1; and generally
we denote by [H] the truth value of formula H.

For quantification we suppose that generalization $\forall$
corresponds to taking the infimum of truth values:

$$[\forall x H(x)] = \inf_{x \in \underline{U}} [H(x)],$$

$\underline{U}$ the common universe of discourse for all our fuzzy sets,
and that existential quantification $\exists$ corresponds to taking
the supremum of truth values.

Obviously, with our notation

$$\Vdash H_1 \rightarrow H_2 \quad \text{iff} \quad [H_1] \leq [H_2] \quad (\mathbf{x})$$

for $\rightarrow$ any one of $\rightarrow_1, \rightarrow_2$. And with biimplication defined
by

$$H_1 \leftrightarrow H_2 =_{\text{def}} (H_1 \rightarrow H_2) \wedge_1 (H_2 \rightarrow H_1),$$

again $\rightarrow$ any one of $\rightarrow_1, \rightarrow_2$ (but the same in both occurrences), we get

$$\Vdash H_1 \leftrightarrow H_2 \quad \text{iff} \quad [H_1] = [H_2]. \quad (+)$$

In the following, sometimes the choice of $\rightarrow_1, \rightarrow_2$ or
also of $\leftrightarrow_1, \leftrightarrow_2$ is inessential. Then we will write $\rightarrow$
resp. $\leftrightarrow$ only. (In all these cases for $\rightarrow, \leftrightarrow$ any connective
may be chosen that meets condition $(\mathbf{x})(\mathbf{+})$ respectively, but
it is not necessary to take this more general point of view
here.)

## 3. Set Theoretical Notation

Generalized membership within a fuzzy set we describe
by our many valued membership predicate $\in$ putting

$$[x \in A] =_{\text{def}} \mu_{A}(x).$$

The fact that a fuzzy set is uniquely characterized by its
many valued language always has truth value 1; and generally we denote by [H] the truth value of formula H.

For quantification we suppose that generalization $\forall$ corresponds to taking the infimum of truth values:

$$[\forall x H(x)] = \inf_{x \in \underline{U}} [H(x)],$$

$\underline{U}$ the common universe of discourse for all our fuzzy sets, and that existential quantification $\exists$ corresponds to taking the supremum of truth values.

Obviously, with our notation

$$\models H_1 \rightarrow H_2 \quad \text{iff} \quad [H_1] \leq [H_2] \quad (\mathbf{x})$$

for $\rightarrow$ any one of $\rightarrow_1, \rightarrow_2$. And with biimplication defined by

$$H_1 \leftrightarrow H_2 =_{\text{def}} (H_1 \rightarrow H_2) \wedge_1 (H_2 \rightarrow H_1),$$

again $\rightarrow$ any one of $\rightarrow_1, \rightarrow_2$ (but the same in both occurrences), we get

$$\models H_1 \leftrightarrow H_2 \quad \text{iff} \quad [H_1] = [H_2]. \quad (+)$$

In the following, sometimes the choice of $\rightarrow_1, \rightarrow_2$ or also of $\leftrightarrow_1, \leftrightarrow_2$ is inessential. Then we will write $\rightarrow resp. \leftrightarrow$ only. (In all these cases for $\rightarrow, \leftrightarrow$ any connective may be chosen that meets condition $(\mathbf{x})(\mathbf{+})$ respectively, but it is not necessary to take this more general point of view here.)

## 3. Set Theoretical Notation

Generalized membership within a fuzzy set we describe by our many valued membership predicate $\in$ putting

$$[x \in A] =_{\text{def}} \mu_A(x).$$

The fact that a fuzzy set is uniquely characterized by its
membership function we use to introduce class terms. Having
given a fuzzy set A and a formula $H(x)$ of our many valued
language such that

$$ \cup_A(x) = [H(x)] \quad \text{for all } x \in \underline{U}, $$

$\underline{U}$ the universe of discourse, we define class terms by

$$ \{x \mid H(x)\} =_{\text{def}} A . $$

The usual intersection of fuzzy sets A,B, denoted: $A \cap_1 B$,
now can be characterized through

$$ A \cap_1 B = \{x \mid x \in A \wedge_1 x \in B\}, $$

the cartesian product e.g. through

$$ A \times_1 B = \{(x,y) \mid x \in A \wedge_1 y \in B\}. $$

More generally, for every binary function $f$ on the universe
of discourse $\underline{U}$ we write ($\wedge$ any one of $\wedge_1, \wedge_2$)

$$ \{f(x,y) \mid x \in A \wedge y \in B\} $$

for the set

$$ \{z \mid \exists x \exists y((x \in A \wedge y \in B) \wedge_1 z \neq f(x,y))\}, $$

putting additionally

$$ [u \doteq v] = \begin{cases} 1, & \text{if } u = v \\ 0, & \text{otherwise.} \end{cases} $$

With this notation, the extension principle of L. ZADEH
/6/ becomes:

Given an operation $*$ within $\underline{U}$, extend it to the fuzzy
subsets of $\underline{U}$ by putting

$$ A * B = \{a * b \mid a \in A \wedge_1 b \in B\}. $$

Finally, we generalize the usual two-valued inclusion
relation for fuzzy sets to many valued ones defining

$$ A \subseteq_i B =_{\text{def}} \forall x (x \in A \rightarrow_i x \in B), $$

with $i=1,2$. Again we write simply $A \subseteq B$ if it does not
membership function we use to introduce class terms. Having
given a fuzzy set A and a formula $H(x)$ of our many valued
language such that

$$ \cup_A(x) = [H(x)] \quad \text{for all } x \in \underline{U}, $$

$\underline{U}$ the universe of discourse, we define class terms by

$$ \{x \mid H(x)\} =_{\text{def}} A. $$

The usual intersection of fuzzy sets A,B, denoted: $A \cap_1 B$,
now can be characterized through

$$ A \cap_1 B = \{x \mid x \in A \wedge_1 x \in B\}, $$

the cartesian product e.g. through

$$ A \times_1 B = \{(x,y) \mid x \in A \wedge_1 y \in B\}. $$

More generally, for every binary function $f$ on the universe
of discourse $\underline{U}$ we write ($\wedge$ any one of $\wedge_1, \wedge_2$)

$$ \{f(x,y) \mid x \in A \wedge y \in B\} $$

for the set

$$ \{z \mid \exists x \exists y((x \in A \wedge y \in B) \wedge_1 z \neq f(x,y))\}, $$

putting additionally

$$ [u \doteq v] = \begin{cases} 1, & \text{if } u = v \\ 0, & \text{otherwise.} \end{cases} $$

With this notation, the extension principle of L. ZADEH
/6/ becomes:

Given an operation $*$ within $\underline{U}$, extend it to the fuzzy
subsets of $\underline{U}$ by putting

$$ A * B = \{a * b \mid a \in A \wedge_1 b \in B\}. $$

Finally, we generalize the usual two-valued inclusion
relation for fuzzy sets to many valued ones defining

$$ A \subseteq_i B =_{\text{def}} \forall x (x \in A \rightarrow_i x \in B), $$

with $i=1,2$. Again we write simply $A \subseteq B$ if it does not
matter if we choose $\mathcal{C}_1$ or $\mathcal{C}_2$. And it is easy to see that

$$ \vdash A \subseteq_2 B \rightarrow A \subseteq_1 B , $$

$$ \vdash A \subseteq_2 B \quad \text{iff} \quad \vdash A \subseteq_1 B , $$

and

$$ \vdash A \subseteq_2 B \wedge_1 B \subseteq_2 C \rightarrow A \subseteq_2 C . $$

### 4. Fuzzy Equations

Now we are ready to describe the results. We follow the way of presentation in /5/. The proofs will be given elsewhere. With some simple facts of our many valued logic, which will not be explained here, they are almost straightforward.

Westart from a binary operation $*$ within our universe of discourse $\mathcal{U}$, and extend it to fuzzy sets over $\mathcal{U}$ in accordance with the extension principle. Immediately we get

**Corollary 1.** For all $x \in \mathcal{U}$

$$ \vdash x \in A * B \leftrightarrow \exists a \exists b (a \in A \wedge_1 b \in B \wedge_1 x \cong a * b) . $$

**Proposition 1.**

$$ (i) \quad \vdash A \subseteq B \rightarrow A * C \subseteq B * C , $$

$$ (ii) \quad \vdash A \subseteq B \rightarrow C * A \subseteq C * B . $$

**Corollary 2.**

$$ \vdash A \subseteq C \wedge_2 B \subseteq D \rightarrow A * B \subseteq C * D . $$

**Proposition 2.**

$$ (i) \quad A * (\mathcal{U}_{j \in J} B_j) = \mathcal{U}_{j \in J} (A * B_j) , $$

$$ (ii) \quad (\mathcal{U}_{j \in J} A_j) * B = \mathcal{U}_{j \in J} (A_j * B) . $$

From this, using the support $|B|$ of fuzzy set B, we get

**Corollary 3.**

$$ A * B = \mathcal{U}_{b \in |B|} (\{x * b \mid x \in A \wedge_1 b \in B\}) . $$
matter if we choose $\mathfrak{C}_1$ or $\mathfrak{C}_2$. And it is easy to see that

$$ \vdash A \subseteq_2 B \rightarrow A \subseteq_1 B , $$

$$ \vdash A \subseteq_2 B \quad \text{iff} \quad \vdash A \subseteq_1 B , $$

and

$$ \vdash A \subseteq_2 B \wedge_1 B \subseteq_2 C \rightarrow A \subseteq_2 C . $$

### 4. Fuzzy Equations

Now we are ready to describe the results. We follow the way of presentation in /5/. The proofs will be given elsewhere. With some simple facts of our many valued logic, which will not be explained here, they are almost straightforward.

Westart from a binary operation $*$ within our universe of discourse $\mathcal{U}$, and extend it to fuzzy sets over $\mathcal{U}$ in accordance with the extension principle. Immediately we get

**Corollary 1.** For all $x \in \mathcal{U}$

$$ \vdash x \in A * B \leftrightarrow \exists a \exists b (a \in A \wedge_1 b \in B \wedge_1 x \cong a * b) . $$

**Proposition 1.**

$$ (i) \quad \vdash A \subseteq B \rightarrow A * C \subseteq B * C , $$

$$ (ii) \quad \vdash A \subseteq B \rightarrow C * A \subseteq C * B . $$

**Corollary 2.**

$$ \vdash A \subseteq C \wedge_2 B \subseteq D \rightarrow A * B \subseteq C * D . $$

**Proposition 2.**

$$ (i) \quad A * (\mathbf{U}_{j \in J} B_j) = \mathbf{U}_{j \in J} (A * B_j) , $$

$$ (ii) \quad (\mathbf{U}_{j \in J} A_j) * B = \mathbf{U}_{j \in J} (A_j * B) . $$

From this, using the support $|B|$ of fuzzy set B, we get

**Corollary 3.**

$$ A * B = \mathbf{U}_{b \in |B|} (\{x * b \mid x \in A \wedge_1 b \in B\}) . $$
**Corollary 4.** If $*$ has in $\mathcal{U}$ an unique left inverse $\nexists$ such that: $z = x*y$ iff $x = z\nexists y$, then

$$A * B = \mathcal{U}_{b \in B_1} (\{z \parallel z \nexists b \in A \wedge_1 b \in B\}) .$$

**Definition 1** (cf. /4/, /5/).

$$C \nexists A =_{\text{def}} \{ y \parallel \forall x (x \in A \to_2 x * y \in C ) \} .$$

**Proposition 3.**

$$\Vdash C \subseteq_2 D \rightarrow C \nexists A \subseteq_2 D \nexists A .$$

**Proposition 4.**

$$(i) \Vdash A * (C \nexists A) \subseteq C ,$$

$$(ii) \Vdash B \subseteq (A * B) \nexists A .$$

**Proposition 5.**

$$(i) \Vdash A * B \subseteq_2 C \rightarrow B \subseteq_2 C \nexists A ,$$

$$(ii) \Vdash B \subseteq C \nexists A \rightarrow A * B \subseteq C .$$

**Corollary 5.**

$$\Vdash A * B \subseteq_2 C \leftrightarrow B \subseteq_2 C \nexists A .$$

**Definition 2.**

$$A \equiv_2 B =_{\text{def}} A \subseteq_2 B \wedge_1 B \subseteq_2 A .$$

**Corollary 6.**

$$\Vdash A \equiv_2 B \quad \text{iff} \quad A = B .$$

**THEOREM.**

$$\Vdash \exists X (A * X \equiv_2 C) \leftrightarrow A * (C \nexists A) \equiv_2 C .$$

**Corollary 7.**

$$(i) \Vdash \exists X (A * X \equiv_2 C) \quad \text{iff} \quad A * (C \nexists A) = C ,$$

$$(ii) \Vdash \exists X (A * X \equiv_2 C) \quad \text{iff} \quad A * X = C \text{ has a solution.}$$

This last result, Corollary 7, contains the main theorem of /5/. The generalization we proved in our theorem now gives
**Corollary 4.** If $*$ has in $\mathcal{U}$ an unique left inverse $\nexists$ such that: $z = x*y$ iff $x = z\nexists y$, then

$$A * B = \mathcal{U}_{b \in B_1} (\{z \parallel z \nexists b \in A \wedge_1 b \in B\}) .$$

**Definition 1** (cf. /4/, /5/).

$$C \nexists A =_{\text{def}} \{ y \parallel \forall x (x \in A \to_2 x * y \in C ) \} .$$

**Proposition 3.**

$$\Vdash C \subseteq_2 D \rightarrow C \nexists A \subseteq_2 D \nexists A .$$

**Proposition 4.**

$$(i) \quad \Vdash A * (C \nexists A) \subseteq C ,$$

$$(ii) \quad \Vdash B \subseteq (A * B) \nexists A .$$

**Proposition 5.**

$$(i) \quad \Vdash A * B \subseteq_2 C \rightarrow B \subseteq_2 C \nexists A ,$$

$$(ii) \quad \Vdash B \subseteq C \nexists A \rightarrow A * B \subseteq C .$$

**Corollary 5.**

$$\Vdash A * B \subseteq_2 C \leftrightarrow B \subseteq_2 C \nexists A .$$

**Definition 2.**

$$A \equiv_2 B =_{\text{def}} A \subseteq_2 B \wedge_1 B \subseteq_2 A .$$

**Corollary 6.**

$$\Vdash A \equiv_2 B \quad \text{iff} \quad A = B .$$

**THEOREM.**

$$\Vdash \exists X (A * X \equiv_2 C) \leftrightarrow A * (C \nexists A) \equiv_2 C .$$

**Corollary 7.**

$$(i) \quad \Vdash \exists X (A * X \equiv_2 C) \quad \text{iff} \quad A * (C \nexists A) = C ,$$

$$(ii) \quad \Vdash \exists X (A * X \equiv_2 C) \quad \text{iff} \quad A * X = C \text{ has a solution.}$$

This last result, Corollary 7, contains the main theorem of /5/. The generalization we proved in our theorem now gives
not only: $A * X = C$ has a solution iff $A * (C \tilde{\wedge} A) = C$,
but furthermore: the degree of equality (in sense of $\equiv_2$)
of $A * (C \tilde{\wedge} A)$ and $C$ is the same as the degree of existence
of a solution of $A * X \equiv_2 C$.

Or, to put it more loosely: if $A * (C \tilde{\wedge} A)$ and $C$ are
nearly equal then $A * X \equiv_2 C$ is (to the same degree) nearly
solvable, and vice versa.

References

/1/ K. GÖDEL, Zum intuitionistischen Aussagenkalkül, Anzeiger Akad. Wiss. Wien, math.-naturwiss. Kl., 69 (1932), 65 - 66.

/2/ S. GOTTWALD, A cumulative system of fuzzy sets, in: Set Theory and Hierarchy Theory (W Marek, M Srebrny, A Zarach eds.), Lect. Notes Math. 537, Springer-Verlag 1976, pp. 109 - 119.

/3/ S. GOTTWALD, Set theory for fuzzy sets of higher level, Fuzzy Sets Systems 2 (1979), 125 - 151.

/4/ E. SANCHEZ, Resolution of composite fuzzy relation equations, Inform. and Control 30 (1976), 38 - 48.

/5/ E. SANCHEZ, Solution of fuzzy equations with extended operations, Memo. UCB/ERL M82/63, Electronics Research Laboratory, University of California, Berkeley, 1982.

/6/ L. ZADEH, The concept of a linguistic variable and its application to approximate reasoning - I, Information Sci. 8 (1975), 199 - 249.
not only: $A * X = C$ has a solution iff $A * (C \tilde{\wedge} A) = C$,
but furthermore: the degree of equality (in sense of $\equiv_2$)
of $A * (C \tilde{\wedge} A)$ and $C$ is the same as the degree of existence
of a solution of $A * X \equiv_2 C$.

Or, to put it more loosely: if $A * (C \tilde{\wedge} A)$ and $C$ are
nearly equal then $A * X \equiv_2 C$ is (to the same degree) nearly
solvable, and vice versa.

References

/1/ K. GÖDEL, Zum intuitionistischen Aussagenkalkül, Anzeiger Akad. Wiss. Wien, math.-naturwiss. Kl., 69 (1932), 65 - 66.

/2/ S. GOTTWALD, A cumulative system of fuzzy sets, in: Set Theory and Hierarchy Theory (W Marek, M Srebrny, A Zarach eds.), Lect. Notes Math. 537, Springer-Verlag 1976, pp. 109 - 119.

/3/ S. GOTTWALD, Set theory for fuzzy sets of higher level, Fuzzy Sets Systems 2 (1979), 125 - 151.

/4/ E. SANCHEZ, Resolution of composite fuzzy relation equations, Inform. and Control 30 (1976), 38 - 48.

/5/ E. SANCHEZ, Solution of fuzzy equations with extended operations, Memo. UCB/ERL M82/63, Electronics Research Laboratory, University of California, Berkeley, 1982.

/6/ L. ZADEH, The concept of a linguistic variable and its application to approximate reasoning - I, Information Sci. 8 (1975), 199 - 249.