Uniqueness results in the representation of families of sets by fuzzy sets

Generalized representation theorem and its application to the construction of fuzzy sets: Existence and uniqueness

“…The sufficient conditions to guarantee O(S • ) = {i S • } have been studied by Jaballah and Saidi [7]. Now, we are going to claim that the sufficient conditions obtained by Jaballah and Saidi [7] are still valid to guarantee O(S) = {i S } when considering S as a subset of (0, 1]. Let S • be a subset of [0, 1] and let sup S • = α * .…”

“…• If S satisfies the sufficient condition obtained by Jaballah and Saidi [7] to guarantee O(S) = {i S }, then we are done.…”

“…We define the family A S = {à α : α ∈ S}. Jaballah and Saidi [7,[15][16][17][18] have studied the problem, which says whether we can construct a fuzzy subsetà of U satisfying R(ξÃ) = S and M S = A S , where R(ξÃ) denotes the rage of ξÃ. However, the 0-level set was also defined as the whole universal set U in Jaballah and Saidi [7,[15][16][17][18].…”

“…Therefore, we shall consider the subset S of (0, 1] in this paper, and try to construct a fuzzy subsetà of U such that R(ξÃ) \ {0} = S. On the other hand, whether the supremum α * = sup S belongs to S or not is an important issue. We are going to construct a fuzzy subsetà of U such that R(ξÃ) \ {0} = S ∪ {α * }, which was not discussed in Jaballah and Saidi [7,[15][16][17][18]. The existence of construction was obtained in Jaballah and Saidi [7, Theorem 1], which was not related the representation theorem as shown in Ralescu [14,Theorem 2].…”

“…In this paper, we shall derive the generalized representation theorem for the purpose of guaranteeing the existence of construction. This is a different methodology for constructing fuzzy sets proposed by Jaballah and Saidi [7].…”