The phrase "a soft set of" is correct and usable in written English.
It can be used when describing a collection or group of items that are gentle, mild, or not harsh in nature.
Example: "She brought a soft set of towels to the spa for a relaxing experience."
Alternatives: "a gentle collection of" or "a mild assortment of".
Example 6.2 Let U = R, E = Z and let F ∈ S ( U ) be a soft set of the form F = { ( p, ( p, p + 1 ] ) : p ∈ E }.
A soft complement of F, denoted by F c ˜, is a soft set of the form F c ˜ = { ( p, U ∖ Λ F ( p ) ) : p ∈ E }.
If ( F, A ) is a soft set of soft topological space X, then sbcl ( ( F, A ) ) is the smallest sb-closed set containing ( F, A ). Thus, sbcl ( F, A ) = ( F, A ) ∪ ~ [ int ( cl ( ( F, A ) ) ) ∩ ~ cl ( int ( ( F, A ) ) ) ].
If ( F, A ) is a soft set of soft topological space X then sbint ( ( F, A ) ) is the largest sb-open set contained in ( F, A ). Thus, sbint ( F, A ) = ( F, A ) ∩ ~ [ int ( cl ( ( F, A ) ) ) ∪ ~ cl ( int ( ( F, A ) ) ) ].
Next, we prove the following theorem which relates the concepts of soft interior and soft closure: Theorem 5. Let F B be a soft set of soft topological space ( F A, τ ~ ) Open image in new window.Then, (1) ( ¯ F B c ~ ) = ( F B ° ) c ~ Open image in new window.
A soft set R ∈ S ( U ) is called a soft relation from F 1 to F 2 if R ⊆ ˜ F 1 × ˜ F 2, i.e., R is a soft set of the form R = { ( ( p, q ), U p × U q ) : p, q ∈ E, U p ⊆ Λ F 1 ( p ), U q ⊆ Λ F 2 ( q ) }.
A softer set of regulations would still afford residents the cachet of living in a "landmark district" without pitting neighbor against neighbor.
It is clear that C n ⊆ ˜ C n − 1 for each n ∈ N. By Proposition 4.2, for each n ∈ N, C n is soft closed and, due to Proposition 6.1, a soft set D of the form D = ⋂ ˜ n ∈ N C n is nonempty.
In this article, firstly, we establish the concept of soft element of a soft set and show its natural connection with the notion of soft set and soft operations.
They defined basic notions and concepts of soft topological spaces such as soft open and closed sets, soft interior, soft closure, soft basis, soft nbd of a point, soft limit point of a soft set, soft complement, soft difference, and soft boundary and established several properties of these soft notions.
In the article first we introduce a new notion of soft element of a soft set and establish its natural relation with soft operations and soft objects in soft topological spaces.
