EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 9, No. 4, 2016, 452-463
ISSN 1307-5543 – www.ejpam.com

Binary Soft Set Theory

Ahu Açıkgöz∗, Nihal Taş
Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey

Abstract. In this paper, we study the concept of soft sets theory which introduced by Molodtsov [9]
and the theory of soft sets which studied by Maji et al. [6]. The authors introduce a binary soft set
on two initial universal sets and a parameter set. Also subset of a binary soft set, soft super set of a
binary soft set, equality of two binary soft sets, complement of a binary soft set, null binary soft set,
absolute binary soft set, union of two binary soft sets, intersection of two binary soft sets, difference
and symmetric difference of two binary soft sets, AND and OR operations via two binary soft sets are
defined by authors in the present paper. Also some properties of binary soft sets are investigated.

2010 Mathematics Subject Classifications: 03E72, 06D72, 97E60
Key Words and Phrases: Soft set, Binary soft set

1. Introduction

Some set theories such as theory of fuzzy sets [13], rough sets [10], intuitionistic fuzzy sets
[3], vague sets [5] etc. can be deal with unclear notions. But, these theories are not sufficient
to solve some difficulties and problems. There are some vague problems in economics, medical
science, social science, finance etc. Then, what is the reason of vague problems and difficulties?
It is possible the insufficiency of the parametrization tool of the theories. In 1999, Molodtsov
[9] introduced the idea of soft set theory as a general mathematical tool for coping with these
difficulties. In 2001, Maji, Biswas and Roy [7] defined the concept of a fuzzy soft set and
[8] an intuitionistic fuzzy soft set. In 2003, Maji et al. [6] studied the theoretical concepts
of the soft set theory. In 2009, Ali et al. [1] investigated several operations on soft sets and
defined some new notions such as the restricted union etc. In 2010, Xu et al. [12] introduced
vague soft sets and studied some properties of them. In 2010, Feng et al. [4] studied soft sets
combined with fuzzy sets and rough sets as a tentative approach. In 2011, Shabir et al. [2]
introduced algebraic structures of soft sets via new notions. In 2011, Naz et al. [11] defined
some notions such as soft topological space, soft interior, soft closure etc.

The aim of this present paper is to introduce the binary soft set on two initial universal
sets and investigated some properties. This paper is organized as below: In Section 2 we

∗Corresponding author.

Email addresses: ahuacikgoz@gmail.com (A. Açıkgöz), nihalarabacioglu@hotmail.com (N. Taş)

http://www.ejpam.com 452 c© 2016 EJPAM All rights reserved.



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 453

give known definitions and a proposition related soft set theory. In Section 3 we introduce
the concept of a binary soft set and the operations on binary soft sets such as binary subset,
binary equality, union of two binary soft sets, intersection of two binary soft sets, difference of
two binary soft sets, symmetric difference of two binary soft sets etc. In Section 4 we give a
characteristic function of binary soft set. The last section is the conclusion.

2. Preliminaries

In this section we recall some definitions and a proposition.

Definition 1 ([9]). A pair (F, A) is called a soft set over X as follows:

F : A→ P(X ).

Definition 2 ([6]). Let (F, A) and (G, B) be two soft sets. Then (F, A) is said to be a soft subset of
(G, B) if A⊆ B and F(e) ⊆ G(e), for all e ∈ A. It is denoted by (F, A)e⊆(G, B). (F, A) is said to be
soft equal to (G, B) if (F, A)e⊆(G, B) and (G, B)e⊆(F, A). It is denoted by (F1, A) = (F2, A).

Definition 3 ([1]). The complement of a soft set (F, A) is defined as (F, A)c = (F c , A), where
F c(e) = (F(e))c = X − F(e), for all e ∈ A.

Definition 4 ([11]). The difference of two soft sets (F, A) and (G, A) is defined by
(F, A)− (G, A) = (F − G, A), where (F − G)(e) = F(e)− G(e), for all e ∈ A.

Definition 5 ([11]). Let (F, A) be a soft set over X and x ∈ X . x is said to be in the soft set (F, E)
denoted by x ∈ (F, A) if x ∈ F(e) for all e ∈ A.

Definition 6 ([6]). A soft set (F,A) over X is said to be a null soft set if F(e) = ;, for all e ∈ A.
This is denoted by e;.

Definition 7 ([6]). A soft set (F, A) over X is said to be an absolute soft set if F(e) = X , for all
e ∈ A. This is denoted by eX .

Definition 8 ([6]). The union of two soft sets (F, A) and (G, B) over the common universe X is
the soft set (H, C), where C = A∪ B and H(e) = F(e) if e ∈ A− B or H(e) = G(e) if e ∈ B − A or
H(e) = F(e)∪ G(e) if e ∈ A∩ B for all e ∈ C.

Definition 9 ([6]). The intersection of two soft sets (F, A) and (G, B) over the common universe
X is the soft set (H, C), where C = A∩ B and for all e ∈ C, H(e) = F(e)∩ G(e).

Definition 10 ([6]). Let E = {ei : 1 ≤ i ≤ n} be a set of parameters. The NOT set of E denoted
by eE = {eei : 1≤ i ≤ n} where eei = not ei for each i.

Proposition 1 ([6]). Let A, B ⊆ E be parameter sets.

(i) e(eA) = A.

(ii) e(A∪ B) =eA∪eB.



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 454

(iii) e(A∩ B) =eA∩eB.

Definition 11 ([6]). Let (F, A) and (G, B) be two soft sets. Then "(F, A)AN D(G, B)" denoted by
(F, A) ∧ (G, B) is defined by (F, A) ∧ (G, B) = (H, A× B), where H(e, f ) = F(e) ∩ G(e) for each
(e, f ) ∈ A× B.

Definition 12 ([6]). Let (F, A) and (G, B) be two soft sets. Then "(F, A)OR(G, B)" denoted by
(F, A) ∨ (G, B) is defined by (F, A) ∨ (G, B) = (O, A× B), where O(e, f ) = F(e) ∪ G(e) for each
(e, f ) ∈ A× B.

3. Binary Soft Sets

Let U1, U2 be two initial universe sets and E be a set of parameters. Let P(U1), P(U2) denote
the power set of U1, U2, respectively. Also, let A, B, C ⊆ E.

Definition 13. A pair (F, A) is said to be a binary soft set over U1, U2, where F is defined as below:

F :A→ P(U1)× P(U2),

F(e) =(X , Y ) for each e ∈ A such that X ⊆ U1, Y ⊆ U2.

Example 1. Consider the following sets:

U1 ={t1, t2, t3, t4, t5} is the set of trousers.

U2 ={b1, b2, b3, b4, b5} is the set of blouses.

E ={e1, e2, e3, e4, e5, e6, e7, e8}.

E is the set of parameters, where e1: expensive, e2: cheap, e3: sport, e4: classic, e5: colorful, e6:
plain, e7: small, e8: large.

The binary soft set (F, A) describes "the special features of both the trousers and the blouses"
which Mrs. X is going to buy, where A= {e1, e2, e3, e4} ⊆ E. (F, A) is a binary soft set over U1, U2
defined as follows:

F(e1) =({t1, t2}, {b1, b3}),
F(e2) =({t3, t4}, {b2, b4, b5}),
F(e3) =({t2, t3, t5}, {b1, b5}),
F(e4) =({t1, t4}, {b2, b3}).

So, we can say the binary soft set

(F, A) ={expensive trousers, blouses : resp. {t1, t2}, {b1, b3};
cheap trousers, blouses : resp. {t3, t4}, {b2, b4, b5};
sport trousers, blouses : resp. {t2, t3, t5}, {b1, b5};
classic trousers, blouses : resp. {t1, t4}, {b2, b3}}.



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 455

We denote the binary soft set (F, A) as below:

(F, A) ={(e1, ({t1, t2}, {b1, b3})), (e2, ({t3, t4}, {b2, b4, b5})), (e3, ({t2, t3, t5}, {b1, b5})),
(e4, ({t1, t4}, {b2, b3}))}.

In this example, we can see the views of Mrs. X who wants to buy both trousers and blouses under
the same parameters.

Definition 14. Let (F, A), (G, B) be two binary soft sets over the common U1, U2. (F, A) is called
a binary soft subset of (G, B) if

(i) A⊆ B,

(ii) X1 ⊆ X2 and Y1 ⊆ Y2 such that F(e) = (X1, Y1), G(e) = (X2, Y2) for each e ∈ A.

We denote it (F, A)ee⊆ (G, B), briefly.
(F, A) is called a binary soft super set of (G, B) if (G, B) is a binary soft subset of (F, A). We

write (F, A)ee⊇ (G, B).

Example 2. Let U1 = {t1, t2, t3, t4, t5}, U2 = {b1, b2, b3, b4, b5} and E = {e1, e2, e3, e4, e5}. Let
A= {e1, e2, e3} ⊆ E and B = {e1, e2, e3, e4} ⊆ E.
(F, A), (G, B) are two binary soft sets over U1, U2 defined as follows:

(F, A) ={(e1, ({t1, t2}, {b1})), (e2, ({t3}, {b3, b4})), (e3, ({t1, t4}, {b1, b2}))},
(G, B) ={(e1, ({t1, t2, t3}, {b1})), (e2, ({t1, t3}, {b3, b4, b5})), (e3, ({t1, t3, t4}, U2)), (e4, (U1, U2))}.

Therefore, (F, A)ee⊆ (G, B).

Definition 15. Let (F, A), (G, B) be two binary soft sets over the common U1, U2. (F, A) is called
a binary soft equal of (G, B) if (F, A) is a binary soft subset of (G, B) and (G, B) is a binary soft
subset of (F, A). We denote it (F, A) = (G, B).

Definition 16. The complement of a binary soft set (F, A) is denoted by (F, A)c and is defined
(F, A)c = (F c , eA), where F c :eA→ P(U1)× P(U2) is a mapping given by F c(e) = (U1−X , U2−Y )
such that F(e) = (X , Y ). Clearly, ((F, A)c)c = (F, A).

Example 3. Consider Example 1. Then

(F, A)c ={not expensive trousers, blouses: resp.{t3, t4, t5}, {b2, b4, b5};
not cheap trousers, blouses: resp.{t1, t2, t5}, {b1, b3};
not sport trousers, blouses: resp.{t1, t4}, {b2, b3, b4};
not classic trousers, blouses: resp.{t2, t3, t5}, {b1, b4, b5}}.

Definition 17. A binary soft set (F, A) over U1, U2 is called a binary null soft set denoted by ee; if
F(e) = (;,;) for each e ∈ A.



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 456

Example 4. Consider the following sets:

U1 ={ j1, j2, j3} is the set of jeans,

U2 ={t1, t2, t3, t4} is the set of t-shirts,

A={e1 = expensive, e2 = smart, e3 = beautiful },

Where A is the set of parameters. Let (F, A) be a binary soft set as follows:

(F, A) = {(e1, (;,;)), (e2, (;,;)), (e3, (;,;))}.

Therefore, (F, A) is a binary null soft set.

Definition 18. A binary soft set (F, A) over U1, U2 is called a binary absolute soft set denoted by
e

eA if F(e) = (U1, U2) for each e ∈ A.

Example 5. Let U1, U2 and A be sets as in Example 4. Let (F, A) be a binary soft set as follows:

(F, A) = {(e1, (U1, U2)), (e2, (U1, U2)), (e3, (U1, U2))}.

Therefore, (F, A) is a binary absolute soft set. Clearly, (eeA)c = ee; and (ee;)c = eeA.

Definition 19. Union of two binary soft sets (F, A) and (G, B) over the common U1, U2 is the
binary soft set (H, C), where C = A∪ B, and for each e ∈ C,

H(e) =











(X1, Y1), e ∈ A− B

(X2, Y2), e ∈ B − A

(X1 ∪ X2, Y1 ∪ Y2), e ∈ A∩ B

such that F(e) = (X1, Y1) for each e ∈ A and G(e) = (X2, Y2) for each e ∈ B. We denote it
(F, A)ee∪ (G, B) = (H, C).

Example 6. Consider the following sets:

U1 ={s1, s2, s3, s4, s5, s6} is the set of shoes,

U2 ={p1, p2, p3, p4} is the set of purses,

E ={e1 = expensive, e2 = cheap, e3 = black,

e4 = brown, e5 = leather, e6 = sport, e7 = classic, e8 = smart}.

Let A = {e1, e3, e5} ⊆ E and B = {e3, e4, e6, e8} ⊆ E. Let (F, A), (G, B) be two binary soft sets as
follows:

(F, A) ={(e1, ({s1, s2}, {p2})), (e3, ({s4, s5, s6}, {p1, p3})), (e5, ({s2, s4, s6}, {p2, p4}))},
(G, B) ={(e3, ({s4, s5}, {p1, p4})), (e4, ({s1}, {p2})), (e6, ({s1, s2}, {p4})), (e8, ({s5}, {p1}))}.

Then (H, C) = (F, A)ee∪ (G, B) is the binary soft set as below such that C = A∪ B:

(H, C) ={(e1, ({s1, s2}, {p2})), (e3, ({s4, s5, s6}, {p1, p3, p4})), (e4, ({s1}, {p2})),
(e5, ({s2, s4, s6}, {p2, p4})), (e6, ({s1, s2}, {p4})), (e8, ({s5}, {p1}))}.



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 457

Definition 20. Intersection of two binary soft sets (F, A) and (G, B) over the common U1, U2
is the binary soft set (H, C), where C = A∩ B, and H(e) = (X1 ∩ X2, Y1 ∩ Y2) for each e ∈ C
such that F(e) = (X1, Y1) for each e ∈ A and G(e) = (X2, Y2) for each e ∈ B. We denote it
(F, A)ee∩ (G, B) = (H, C).

Example 7. In the Example 6, intersection of two binary soft sets (F, A) and (G, B) is the binary
soft set (H, C), where C = A∩ B = {e3} and (H, C) = {(e3, ({s4, s5}, {p1}))}.

Proposition 2. Let (F, A), (G, B) and (H, C) be three binary soft sets. Then we have the following
results:

(i) (F, A)ee∪ (F, A) = (F, A).

(ii) (F, A)ee∪ (G, B) = (G, B)ee∪ (F, A).

(iii) (F, A)ee∪ ((G, B)ee∪ (H, C)) = ((F, A)ee∪ (G, B))ee∪ (H, C).

(iv) (F, A)ee∪ee;= (F, A).

(v) (F, A)ee∪ eeA= eeA.

(vi) (F, A)ee⊆ (F, A)ee∪ (G, B) and (G, B)ee⊆ (F, A)ee∪ (G, B).

(vii) (F, A)ee∪ (G, B) = ee; if and only if (F, A) = ee; and (G, B) = ee;.

(viii) (F, A)ee⊆ (G, B) if and only if (F, A)ee∪ (G, B) = (G, B).

Proof. It is obvious from Definition 19.

Proposition 3. Let (F, A), (G, B) and (H, C) be three binary soft sets. Then we have the following
results:

(i) (F, A)ee∩ (F, A) = (F, A).

(ii) (F, A)ee∩ (G, B) = (G, B)ee∩ (F, A).

(iii) (F, A)ee∩ ((G, B)ee∩ (H, C)) = ((F, A)ee∩ (G, B))ee∩ (H, C).

(iv) (F, A)ee∩ee;= ee;.

(v) (F, A)ee∩ eeA= (F, A).

(vi) (F, A)ee∩ (G, B)ee⊆ (F, A) and (F, A)ee∪ (G, B)ee⊆ (G, B).

(vii) (F, A)ee∩ (G, B) = ee; if and only if (F, A) = ee; or (G, B) = ee;.

(viii) (F, A)ee⊆ (G, B) if and only if (F, A)ee∩ (G, B) = (F, A).



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 458

Proof. It is obvious from Definition 20.

Proposition 4. Let (F, A) and (G, B) be two binary soft sets. Then we have the following results:

(i) (F, A)ee∪ (F, A)c = eeA.

(ii) (F, A)ee∩ (F, A)c = ee;.

(iii) (F, A)ee⊆ (G, B) if and only if (G, B)cee⊆ (F, A)c .

(iv) ((F, A)ee∪ (G, B))c = (F, A)cee∪ (G, B)c .

(v) ((F, A)ee∩ (G, B))c = (F, A)cee∩ (G, B)c .

Proof.

(i) It is obvious from Definitions 16, 19, and 20.

(ii) It is obvious from Definitions 16, 19, and 20.

(iii) It is obvious from Definitions 16, 19, and 20.

(iv) Let (F, A)ee∪ (G, B) = (H, A∪ B), where for each e ∈ A∪ B

H(e) =











(X1, Y1), e ∈ A− B

(X2, Y2), e ∈ B − A

(X1 ∪ X2, Y1 ∪ Y2), e ∈ A∩ B

such that F(e) = (X1, Y1) for each e ∈ A and G(e) = (X2, Y2) for each e ∈ B. Hence,
((F, A)ee∪ (G, B))c = (H, A∪ B)c = (H c , eA∪eB).

Now, H c(ee) = (U1 − X , U2 − Y ) for each ee ∈eA∪eB such that H(e) = (X , Y ). Therefore,

H c(ee) =











(U1 − X1, U2 − Y1), ee ∈eA−eB
(U1 − X2, U2 − Y2), ee ∈eB−eA
(U1 − (X1 ∪ X2), U2 − (Y1 ∪ Y2)), ee ∈eA∩eB

(1)

Now, (F, A)cee∪ (G, B)c = (F c , eA)ee∪ (Gc , eB) = (K , eA∪eB), where

K(ee) =











(U1 − X1, U2 − Y1), ee ∈eA−eB
(U1 − X2, U2 − Y2), ee ∈eB−eA
(U1 − (X1 ∪ X2), U2 − (Y1 ∪ Y2)), ee ∈eA∩eB

. (2)

Finally, H c and K are same. So, proof is completed.

(v) It is proved by a similar way.



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 459

Proposition 5. Let (F, A), (G, B) and (H, C) be three binary soft sets. Then we have the following
results:

(i) (F, A)ee∪ ((G, B)ee∩ (H, C)) = ((F, A)ee∪ (G, B)) ee∩ ((F, A)ee∪ (H, C)).

(ii) (F, A)ee∩ ((G, B)ee∪ (H, C)) = ((F, A)ee∩ (G, B)) ee∪ ((F, A)ee∩ (H, C)).

Proof. It is obvious from Definitions 7 and 8.

Definition 21. The difference of two binary soft sets (F, A) and (G, A) over the common U1, U2 is
the binary soft set (H, A), where H(e) = (X1−X2, Y1−Y2) for each e ∈ A such that (F, A) = (X1, Y1)
and (G, A) = (X2, Y2).

Example 8. Consider the following sets:

U1 ={c1, c2, c3, c4, c5} is the set of computers.

U2 ={m1, m2, m3, m4, m5} is the set of mobile phones.

E ={e1 = expensive, e2 = outlook, e3 = functions }.

Let (F, E), (G, E) be two binary soft sets as follows:

(F, E) ={(e1, ({c1, c3}, {m2, m3})), (e2, ({c4}, {m1, m5})), (e3, ({c3, c4}, {m2}))}.
(G, E) ={(e1, ({c1, c4}, {m1})), (e2, ({c4}, {m2, m5})), (e3, ({c4}, {m2}))}.

Then (H, E) = {(e1, ({c3}, {m2, m3})), (e2, (;, {m1})), (e3, ({c3},;))}.

Definition 22. The symmetric difference of two binary soft sets (F, A) and (G, A) over the common
U1, U2 is the binary soft set (H, A) defined (H, A) = ((F, A)− (G, A))ee∪ ((G, A)− (F, A)). We denote
it (H, A) = (F, A)∆(G, A).

Example 9. In the Example 8, symmetric difference of two binary soft sets (F, E) and (G, E) is
the binary soft set (H, E) as follows:

(H, E) = {(e1, ({c3, c4}, {m1, m2, m3})), (e2, (;, {m1, m2})), (e3, ({c3},;))}.

Proposition 6. Let (F, A), (G, A) and (H, A) be three binary soft sets. Then we have the following
results:

(i) eeA−ee;= eeA and eeA− eeA= ee;.

(ii) eeA− (F, A)c = (F, A).

(iii) (F, A)ee⊆ (G, A) if and only if (G, A)c ee⊆ (F, A)c .

(iv) (F, A)ee∩ (G, A) if and only if (F, A)ee⊆ (G, A)c if and only if (G, A)ee⊆ (F, A)c .



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 460

(v) (F, A)ee∪ (G, A) = eeA, (F, A)ee∩ (G, A) = ee; if and only if (F, A) = (G, A)c .

(vi) ((F, A)ee∪ (G, A))c = (F, A)c ee∩ (G, A)c .

(vii) ((F, A)ee∩ (G, A))c = (F, A)c ee∪ (G, A)c .

(viii) If (F, A)ee⊆ (G, A), (F, A)ee∪ (H, A) ee⊆ (G, A)ee∪ (H, A).

(ix) If (F, A)ee⊆ (G, A), (F, A)ee∩ (H, A) ee⊆ (G, A)ee∩ (H, A).

(x) If (F, A)ee⊆ (G, A) and (F, A)ee⊆ (H, A), (F, A)ee⊆ (G, A)ee∩ (H, A).

(xi) If (F, A)ee⊆ (G, A) and (F, A)ee⊆ (H, A), (F, A)ee∪ (G, A)ee⊆ (H, A).

(xii) (F, A)− ((G, A)− (H, A)) = (F, A)− ((G, A)ee∪ (H, A)).

(xiii) (F, A)− ((G, A)ee∩ (H, A)) = ((F, A)− (G, A)) ee∪ (F, A)− (H, A)).

(xiv) (F, A)∆ee;= (F, A), (F, A)∆(F, A) = ee;, (F, A)∆(G, A) = (G, A)∆(F, A).

(xv) (F, A)∆((G, A)∆(H, A)) = ((F, A)∆(G, A))∆(H, A).

(xvi) (F, A)ee∩ ((G, A)∆(H, A)) = ((F, A)ee∩ (G, A)) ∆ ((F, A)ee∩ (H, A)).

Proof. It is obvious from Definitions 16, 19, 20, 21, and 22.

Definition 23. If (F, A) and (G, B) are two binary soft sets then "(F, A)AN D(G, B)" denoted by
(F, A)ee∧ (G, B) is defined by (F, A)ee∧ (G, B) = (H, A× B), where H(e, f ) = (X1 ∩ X2, Y1 ∩ Y2) for
each (e, f ) ∈ A× B such that F(e) = (X1, Y1) and G(e) = (X2, Y2).

Example 10. In the Example 6, (H, A× B) = (F, A)ee∧ (G, B) is the binary soft set as follows:

(H, A× B) ={((e1, e3), (;,;)), ((e1, e4), ({s1}, {p2})), ((e1, e6), ({s1, s2},;)), ((e1, e8), (;,;)),
((e3, e3), ({s4, s5}, {p1})), ((e3, e4), (;,;)), ((e3, e6), (;,;)), ((e3, e8), ({s5}, {p1})),
((e5, e3), ({s4}, {p4})), ((e5, e4), (;, {p2})), ((e5, e6), ({s2}, {p4})), ((e5, e8), (;,;))}.

Definition 24. If (F, A) and (G, B) are two binary soft sets then "(F, A)OR(G, B)" denoted by
(F, A)ee∨ (G, B) is defined by (F, A)ee∨ (G, B) = (O, A× B), where O(e, f ) = (X1 ∪ X2, Y1 ∪ Y2) for
each (e, f ) ∈ A× B such that F(e) = (X1, Y1) and G(e) = (X2, Y2).

Example 11. In the Example 6, (O, A× B) = (F, A)ee∨ (G, B) is the binary soft set as follows:

(O, A× B) ={((e1, e3), ({s1, s2, s4, s5}, {p1, p2, p4})), ((e1, e4), ({s1, s2}, {p1, p2})),
((e1, e6), ({s1, s2}, {p2, p4})), ((e1, e8), ({s1, s2, s5}, {p1, p2})), ((e3, e3),

({s4, s5, s6}, {p1, p3, p4})), ((e3, e4), ({s1, s4, s5, s6}, {p1, p2, p3})), ((e3, e6),



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 461

({s1, s2, s4, s5, s6}, {p1, p3, p4})), ((e3, e8), ({s4, s5, s6}, {p1, p3})), ((e5, e3),

({s2, s4, s5, s6}, {p1, p2, p4})), ((e5, e4), ({s1, s2, s4, s6}, {p2, p4})), ((e5, e6),

({s1, s2, s4, s6}, {p2, p4})), ((e5, e8), ({s2, s4, s5, s6}, {p1, p2, p4}))}.

Proposition 7. Let (F, A), (G, B) and (H, C) be three binary soft sets. Then we have the following
results:

(i) ((F, A)ee∨ (G, B))c = (F, A)c ee∧ (G, B)c .

(ii) ((F, A)ee∧ (G, B))c = (F, A)c ee∨ (G, B)c .

(iii) ((F, A)ee∨ (G, B))ee∨ (H, C) = (F, A)ee∨ ((G, B)ee∨ (H, C)).

(iv) ((F, A)ee∧ (G, B))ee∧ (H, C) = (F, A)ee∧ ((G, B)ee∧ (H, C)).

(v) (F, A)ee∨ ((G, B)ee∧ (H, C) = ((F, A)ee∨ (G, B)) ee∧ ((F, A)ee∨ (H, C)).

(vi) (F, A)ee∧ ((G, B)ee∨ (H, C) = ((F, A)ee∧ (G, B)) ee∨ ((F, A)ee∧ (H, C)).

Proof. It is obvious from Definitions 23 and 24.

4. A Characteristic Function of The Binary Soft Set

In this section, we give a characteristic function of the binary soft set. We can easily see
elements of initial universal sets providing parameter properties. The characteristic function
is as follows:

Consider the following sets:

U1 ={h j : 1≤ j ≤ n},
U2 ={tk : 1≤ k ≤ m},
E ={ei : 1≤ i ≤ p}.

Let (F, E) = {(ei , (X i , Yi)) : 1≤ i ≤ p, X i ⊆ U1, Yi ⊆ U2} is the binary soft set.
The characteristic functions of (F, E) binary soft set as below:

fE(ei) j =

¨

1, h j ∈ X i

0, h j /∈ X i

and

gE(ei)k =

¨

1, tk ∈ Yi

0, tk /∈ Yi
.

We can form a table which shows the elements providing parameters properties.



A. Açıkgöz, N. Taş / Eur. J. Pure Appl. Math, 9 (2016), 452-463 462

Table 1: The Elements of Initial Universal Sets Providing Parameter Properties.

E/U1 − U2 h1 h2 . . . hn t1 t2 . . . tm

e1 fE(e1)1 fE(e1)2 . . . fE(e1)n gE(e1)1 gE(e1)2 . . . gE(e1)m
e2 fE(e2)1 fE(e2)2 . . . fE(e2)n gE(e2)1 gE(e2)2 . . . gE(e2)m

e1
...

... . . .
...

...
... . . .

...
ep fE(ep)1 fE(ep)2 . . . fE(ep)n gE(ep)1 gE(ep)2 . . . gE(ep)m

Now we investigate a following example:

Example 12. Consider the following sets:

U1 ={h1, h2, h3} is the set of houses,

U2 ={c1, c2, c3, c4} is the set of cars,

E ={e1 = expensive, e2 = sport, e3 = beautiful, e4 = cheap }

where E is the set of parameters. Let (F, E) is a binary soft sets as follows:

(F, E) ={(e1, (X1 = {h1}, Y1 = {c2})), (e2, (X2 = ;, Y2 = {c2, c3})), (e3, (X3 = {h1, h3},
Y3 = {c2, c4})), (e4, (X4 = {h2}, Y4 = {c1}))}.

Then, for 1≤ i ≤ 4, 1≤ j ≤ 3 and 1≤ k ≤ 4,

fE(ei) j =

¨

1, h j ∈ X i

0, h j /∈ X i

and

gE(ei)k =

¨

1, ck ∈ Yi

0, ck /∈ Yi
.

Table 2: Houses and Cars Providing Parameter Properties. In the above table, we see some houses and cars
providing parameter properties. For example, the first house is expensive but the second and third houses
are not expensive. Similarly, the first car is cheap but the other cars are not cheap.

E/U1 − U2 h1 h2 h3 c1 c2 c3 c4

e1 1 0 0 0 1 0 0
e2 0 0 0 0 1 1 0
e3 1 0 1 0 1 0 1
e4 0 1 0 1 0 0 0

5. Conclusion

In this paper, we give a definition of binary soft set on two initial universal sets and a
parameter set. Also we define the binary soft subset, the binary soft equality, the binary null



REFERENCES 463

soft set, the binary absolute soft set, the union of two binary soft sets, the intersection of
two binary soft sets, the difference of two binary soft sets, the symmetric difference of two
binary soft sets, AND operation and OR operation via two binary soft sets. We investigate some
properties of binary soft sets with defined operations (union, intersection etc.) Definition of a
soft set can be given on n - dimension initial universal sets and a parameter set as follows:

F : A→
n
∏

i=1

P(Ui)

where Ui are initial universal sets for 1≤ i ≤ n and A is a parameter set.

References

[1] M.I. Ali, F. Feng, X. Liu, W.K. Min, and M. Shabir. On some new operations in soft set theory,
Computers and Mathematics with Applications, 57, 1547-1553. 2009.

[2] M. Ali, M. Shabir, and M. Naz. Algebraic structures of soft sets associated with new opera-
tions, Computers and Mathematics with Applications, 61, 2647-2654. 2011.

[3] K. Atanassov. Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96. 1986.

[4] F. Feng, C. Li, B. Davraz, and M. Ali. Soft sets combined with fuzzy sets and rough sets: a
tentative approach, Soft Computing, 14, 899 - 911. 2010.

[5] W.L. Gau and D.J. Buehrer. Vague sets, IEEE Transactions System Man Cybernet, 23(2),
610-614. 1993.

[6] P.K. Maji, R. Biswas, and A.R. Roy. Soft set theory, Computers and Mathematics with
Applications, 45, 555-562. 2003.

[7] P.K. Maji, R. Biswas, and A.R. Roy. Fuzzy soft sets, Journal of Fuzzy Mathematics, 9, 589-
602. 2001.

[8] P.K. Maji, R. Biswas, and A.R. Roy. Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathe-
matics, 9, 677-692. 2001.

[9] D. Molodtsov. Soft set theory - First results, Computers and Mathematics with Applica-
tions, 37, 19-31. 1999.

[10] Z. Pawlak. Rough sets, International Journal of Computer and Information Sciences, 11,
341-356. 1982.

[11] M. Shabir and M. Naz. On soft topological spaces, Computers and Mathematics with Ap-
plications, 61, 1786-1799. 2011.

[12] W. Xu, J. Ma, S. Wang, and G. Hao. Vague soft sets and their properties, Computers and
Mathematics with Applications, 59, 787-794. 2010.

[13] L.A. Zadeh. Fuzzy sets, Information and Control, 8, 338-353. 1965.