Erratum to “A note on soft sets, rough soft sets and fuzzy soft sets” [Appl. Soft Comput. 11 (2011) 3329–3332]

Applied Soft Computing 11 (2011) 5817–5818

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Erratum

Erratum to “A note on soft sets, rough soft sets and fuzzy soft sets” [Appl. Soft Comput. 11 (2011) 3329–3332] Yong Yang ∗ , Chenli Ji College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China

a r t i c l e

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Article history: Received 21 June 2011 Accepted 17 July 2011 Available online 24 July 2011

Errors have been found in Example 3 in Section 5 in the above article where: If he considers p as the sole decision parameter then we have a fuzzy subset of P(U) as the following:



Dp =

1

1

1

0 0 0 0 1 3 , , 2 , 2 , , , , ∅ {a} {b} {d} {a, b} {b, d} {a, d} {a, b, d}



.

This fuzzy subset give rise to equivalence relation ı(p) whose equivalence classes are {∅ , {a}, {d}, {a, d}}, {{b}}, {{a, b}, {b, d}}, {{a, b, d}}. Order among these classes is given by {{b}} > {{a, b}, {b, d}} > {{a, b, d}} > {∅ , {a}, {d}, {a, d}}. If he considers c as the sole decision parameter then we have a fuzzy subset of P(U) as the following:



Dc =

1

1

1

1

1

0 2 1 0 3 , 2 , 2 , 2 , , , , ∅ {a} {b} {d} {a, b} {b, d} {a, d} {a, b, d}



.

This fuzzy subset give rise to equivalence relation ı(c) whose equivalence classes are {∅ , {b}}, {{a}, {d}, {a, b}, {b, d}}, {{a, d}}, {{a, b, d}}. Order among these classes is given by {{a, d}} > {{a}, {d}, {a, b}, {b, d}} > {{a, b, d}} > {∅ , {b}}. However, Dp and Dc do not satisfy the definition of the fuzzy subset given in [1]:  Let (F, A) be a soft set over a set U. Then F : A → P(U) is a mapping. For all a ∈ A define a map Da : P(U) → [0, 1] such that Da (X) = |F(a) ∩ X| if F(a) = / ∅ |F(a)| 0 if F(a) = ∅ Then clearly Da is a fuzzy set over P(U) for each a ∈ A. Therefore (F, A) induces a fuzzy soft set over P(U). The following result is obvious. Proposition.

Let (F, A) be a soft set over a set U. For any a ∈ A , if |F(a) | = n , then the partition P(U)/ı(a) has n + 1 equivalence classes at most.

In Example 3 in [1], the partitions P(U)/ı(p) and P(U)/ı(c) have four equivalence classes, which contradict the facts that |F(p) | = 1 and |F(c) | = 2. p as the sole decision parameter, then F(p) = {b} and |F(p) | = 1, we have a fuzzy subset Dp Dp =  0If 0he 1 considers 0 1 1 0 1 , , , , , ∅ {a} {b} {d} {a,b} {b,d} , {a,d} , {a,b,d} . The two equivalence classes of the equivalence relation ı(p) are {∅ , {a}, {d}, {a, d}}, {{b}, {a, b}, {b, d}, {a, b, d}}. Order among these classes is given by {{b}, {a, b}, {b, d}, {a, b, d}} > {∅ , {a}, {d}, {a, d}}. If he considers c as the sole decision parameter, then F(c) = {a, d} and |F(c) | = 2, we have a fuzzy subset Dc



Dc =

1

1

1

1

0 0 2 1 1 , , , 2 , 2 , 2 , , ∅ {a} {b} {d} {a, b} {b, d} {a, d} {a, b, d}

DOI of original article: 10.1016/j.asoc.2011.01.003. ∗ Corresponding author. E-mail address: [email protected] (Y. Yang). 1568-4946/$ – see front matter doi:10.1016/j.asoc.2011.07.008



.

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Y. Yang, C. Ji / Applied Soft Computing 11 (2011) 5817–5818

The three equivalence classes of the equivalence relation ı(c) are {∅ , {b}}, {{a}, {d}, {a, b}, {b, d}}, {{a, d}, {a, b, d}}. Order among these classes is given by {{a, d}, {a, b, d}} > {{a}, {d}, {a, b}, {b, d}} > {∅ , {b}}. Reference [1] Muhammad Irfan Ali, A note on soft sets, rough soft sets and fuzzy soft sets, Applied Soft Computing 11 (2011) 3329–3332.