4.2. Limits of -Soft Sets
Let
,
. Denote:
Then, is called the neighborhood of , is said to be the neighborhood of having no heart, is the center of the neighborhood and is the radius of the neighborhood.
is referred to as the right neighborhood of ,
is said to be the left neighborhood of .
Obviously, .
Given that
is an
-soft set over
U, for
,
, denote:
Remark 1.
Definition 15. Let be an -soft set over U. For , define:
which is called the over-right limit of as ;
which is said to be the under-right limit of as .
which is referred to as the over-left limit of as .
which is said to be the under-left limit of as .
The following theorem shows that the limits can be characterized by and .
Theorem 3. Suppose that is an -soft set over U. Then, for ,
.
.
.
.
.
Proof. Obviously,
. We only need to prove
. Suppose
. Then,
. Pick
. We have
. Therefore,
,
is finite. Denote:
Put
. Then:
Therefore, . However, . This is a contradiction. Thus, .
Put:
Obviously, . We only need to prove . Suppose . Then, . Pick . Then, .
Claim is infinite.
In fact, suppose that
is finite. Put:
Then, , . Therefore, , but . This is a contradiction.
Since , is infinite, we have . However, . This is a contradiction. Thus, .
The proof is similar to (1).
The proof is similar to (2). ☐
Example 7. Consider Example 2, and pick . We have: Lemma 1. Given that is an -soft set over U, then, for ,
.
.
.
.
Proof. To prove
, it suffices to show that:
. Let , . Put . Then,
Since , by Theorem 3(1), we have pick . Then, .
This implies . Thus, .
. , pick .
By the condition, Then, . Thus, , .
By Theorem 3(1), .
By (1) and Theorem 3(2),
.
Hence, .
The proof is similar to (1).
The proof is similar to (2). ☐
Lemma 2. Let be an -soft set over U. Then, for ,
.
.
.
.
Proof. Put . Then, . Therefore, Thus, .
Put . Then, . Therefore,
Thus, .
It is similar to the proof of (1).
It is similar to the proof of (2). ☐
Theorem 4. Suppose that is an -soft set over U. Then, for ,
; if is increasing, then: ; if is decreasing, then: ; if is decreasing, then: ; if is increasing, then: Proof. This holds by Lemmas 1 and 2. ☐
Definition 16. Given that is an -soft set over U, then, for ,
If , then is said to have the limit S as (or has the right-limit S as ), which is denoted by , i.e., ;
if , then is said to have no limit as (or has no right-limit as ).
If , then is said to have the limit S as (or has the left-limit S as ), which is denoted by , i.e., ;
if , then is said to have no limit as (or has no left-limit as ).
If , then is said to have the limit S as , which is denoted by , i.e., ;
if , then is said to have no limit as .
Definition 17. Let be an -soft set over U. Then, for ,
If , then is said to have the over-limit S as , which is denoted by , i.e., ;
if , then is said to have no over-limit as .
If , then is said to have the under-limit S as , which is denoted by , i.e., ;
if , then is said to have no under-limit as .
If , then is said to have the limit as , which is denoted by , i.e., ;
if , then is said to have no limit as .
Remark 2. The limit in Definition 16(3) and the limit in Definition 17(3) are consistent.
Example 8. Let be a constant -soft set over U where . Then, for , .
Obviously, , .
Then,
Similarly,
Thus,
Other types of limits of -soft sets are proposed by the following definition, and these limits can be discussed in a similar way.
Definition 18. Let be an -soft set over U. Define: 4.3. Properties of Limits of -Soft Sets
Proposition 4. For the over-right limit, the following properties hold:
If , then .
.
.
If , then , .
;
.
Proof.
, by Theorem 3(1), . Pick . Then, , .
1) If , then . By the condition, . Then, . This implies . Therefore, .
2) If , then . Therefore, . Since , we have .
By 1) and 2), . By Theorem 3(1), .
. This holds by (1).
. Suppose
. Then:
Pick . We have:
and .
By Theorem 3,
Pick
= min
. Then,
and
. It follows that:
By Remark 1,
Thus, This is a contradiction.
. Then,
. By Theorem 3,
,
. By Remark 1,
. Thus,
Conversely, the proof is similar.
Suppose that , or .
1) If , then . Pick .
Since . Then, . Therefore,
Thus, . This is a contradiction.
2) If , then . Therefore, .
Since
, we have
,
. Therefore,
This is a contradiction.
, we have
. Since:
we have
,
,
. It follows that
,
. Then,
and
. Therefore,
Thus, .
2)
, we have:
Then,
,
,
,
. Then,
. Therefore,
Conversely, the proof is similar.
Proposition 5. For the under-right limit, the following properties hold.
If , then .
.
.
If , then , .
.
Proof. It is similar to the proof of Proposition 4(1).
. This holds by (1).
. Suppose . Then, Pick . We have:
, and .
Pick
= min
. Then,
. It follows that:
By Remark 1,
Thus, This is a contradiction.
. Then, . By Theorem 3, , . By Remark 1, .
Thus,
Conversely, the proof is similar.
By Proposition 4(3),
Since , we have
By Proposition 4(4), ,
, by Theorem 4(2),
Then,
,
,
. It follows that
,
. Then,
By Theorem 4(2), , . Thus, .
, By Theorem 4(2),
Then, , , , , .
Put
. Then,
,
. Then,
,
,
. It follows that
. Therefore,
By Theorem 4(2), .
Proposition 6. For the over-left limit, the following properties hold:
If , then .
.
.
If , then , .
.
.
Proof. The proof is similar to Proposition 4. ☐
Proposition 7. For the under-left limit, the following properties hold:
If , then .
.
.
If , then , .
.
Proof. The proof is similar to Proposition 5. ☐
Corollary 1. Suppose that is an -soft set over U and . For ,
If or , then: If or , then: Proof. This holds by Propositions 4, 5, 6 and 7. ☐
Corollary 2. Given that is an -soft set over U and , for ,
If or , then: If or , then: Proof. This follows from Propositions 4, 5, 6 and 7. ☐
Theorem 5. For the over limit, the following properties hold:
If , then .
.
.
If , then , .
.
Proof. This is a direct result from Propositions 4 and 6. ☐
Theorem 6. For the under limit, the following properties hold:
If , then .
.
.
If , then , .
.
Proof. This holds by Propositions 5 and 7. ☐
Lemma 3. Let be an -soft set over U. For , denote: Proof. Suppose . Then, .
Pick
. Then,
. Therefore,
,
Put . Then, . It follows that . Then, . This is a contradiction.
Thus, .
On the other hand, suppose ; we have .
Pick . Then, . Therefore, . This implies . Then, . Therefore, . This is a contradiction.
Thus, .
Hence, . ☐
Theorem 7. Suppose that is an -soft set over U. Then, for ,
.
.
Proof. Similar to the proof of Theorem 3(1), we have:
.
By Lemma 3,
.
Similar to the proof of Theorem 3(2), we have:
.
By Proposition 4(3), .
By Proposition 6(3), .
By (1),
☐
Theorem 8. Given that is an -soft set over U, then, for ,
.
.
Proof. This follows from Theorem 7. ☐
Theorem 9. For the right limit, the following properties hold:
If , then .
If , , then , .
.
Proof. This holds by Propositions 4 and 5. ☐
Theorem 10. For the left limit, the following properties hold:
If , then .
If , , then , .
.
Proof. This holds by Propositions 6 and 7. ☐
Theorem 11. For the limit, the following properties hold:
If , then .
If , , then , .
.
Proof. This follows from Theorems 9 and 10. ☐