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 }.
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 }.
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 ) ) ) ].
Let ( F, K ) be a soft set of Y, and ( F, A ) be a soft open set of X, such that f - 1 ( ( F, K ) ) ⊂ ~ ( F, A ). Then ( G, K ) = ( f ( ( F, A ) c ) ) c is a sb-open set in Y such that ( F, K ) ⊂ ~ ( G, K ) and f - 1 ( ( G, K ) ) ⊂ ~ ( F, A ). Conversely, suppose that ( F, B ) is a soft closed set of X.
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.
We define on U a soft set F A as follows: F A = { ( small, { n ∈ U : n < 100 } ), ( very big, { n ∈ U : n > 10 6 } ) }. Obviously, an element ( big, ∅ ) is also the element of F A. Definition 2.2 An empty soft set, denoted by ∅ ˜, is a soft set of the form { ( p, ∅ ) : p ∈ E }. Example 2.2 Let U = { white, yellow, red, azure }, E = { dark, ugly }.
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 ) }.
British folk artist Ben Howard played a soft set of songs from his "Every Kingdom" album, while the Dropkick Murphys got the audience dancing with their crowd-rousing Irish rock tracks including "Rose Tattoo".
Another monitor tuned into a lower frequency detects a much softer set of clicks.
