Information Sciences 166 (2004) 167–179

www.elsevier.com/locate/ins

Some more properties of lists and fuzzy lists

B.K. Tripathy a,*, S.S. Gantayat b

a Department of Computer Science, Berhampur University, Berhampur 760 007, Orissa, Indiab Hi-Tech Institute of Information Technology, Jeypore 764 002, Orissa, India

Received 20 February 2003; received in revised form 20 June 2003; accepted 20 October 2003

Abstract

In this article we introduce the map operator on a list, define list indexing the cons

operator and establish their properties. Parallel notions and properties for fuzzy lists are

also considered.

� 2003 Elsevier Inc. All rights reserved.

Keywords: List; Fuzzy list; Take; Drop; Map; Indexing; Cons

1. Introduction

The concept of lists [1] is a generalization of the notion of bags [6], which is

itself a generalization of the concept of sets. In fact the number of occurrences

of an element and their order of occurrence is significant in a list, which areunimportant in the setting of sets.

Jena et al. redefined the notion of list very recently [2]. Some of the well-

known operations on lists were also defined from the new direction and

properties of these operations were established. This study was further carried

out by Tripathy and Pattnaik [4,5] by the introduction and study of the

operators ‘take’ and ‘drop’ on a list.

In this paper we establish one more property of the ‘take’ and ‘drop’

operator; define the notions of ‘map’, ‘list indexing’, ‘interval of natural

* Corresponding author.

E-mail addresses: tripathybk@rediffmail.com (B.K. Tripathy), sgantayat67@rediffmail.com

(S.S. Gantayat).

0020-0255/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2003.10.014

168 B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179

numbers’, ‘sublist’, the ‘cons’ operator and prove some theorems which

establish properties of these notions.

2. Definitions

Definition 2.1. A list (crisp list) L drawn from a set X is represented by a

position function PL defined as

PL : X ! P ðNÞ;

where P ðNÞ denotes power set of non-negative integers N .

Definition 2.2. For any finite list L drawn from X , the cardinality of L isdenoted by #L and is defined as

#L ¼Xx2X

jPLðxÞj;

whenever the right hand side exists. In this case L is said to be finite. Otherwise,

L is said to be infinite.

Definition 2.3. For any finite list L drawn from X , the reverse of L is denoted by

revðLÞ on X , and is given by the position function

PrevðLÞðxÞ ¼ f#L� t � 1 : t 2 PLðxÞg:

Definition 2.4. For any finite list L drawn from X , the head of L is denotedby hdðLÞ, where

hdðLÞ ¼ x; if 0 2 PLðxÞ:

Definition 2.5. For any finite list L drawn from X , the last of L is denoted by

lastðLÞ, where

lastðLÞ ¼ x; if ð#L� 1Þ 2 PLðxÞ:

Definition 2.6. A list L is empty if PLðxÞ ¼ u for each x 2 X and we denote as

L ¼ ½ �.

Definition 2.7. Let L1 and L2 be two finite lists drawn from X . Then we define

the concatenation of L1 and L2 denoted by L1,L2 and is given by the position

function

PL1#L2ðxÞ ¼ PL1ðX Þ [ f#L1 þ t : t 2 PL2ðxÞg; 8x 2 X :

B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179 169

Definition 2.8. Let L be list drawn from the set X , then we define the take

operator on L such that for any n 2 N , takeðn;LÞ is a list on X given by the

position function

Ptakeðn;LÞðxÞ ¼u; if n6 min PLðxÞ or PLðxÞ ¼ u;fr : r < n ^ r 2 PLðxÞg; if n > min PLðxÞ

�

for any x 2 X .

Definition 2.9. Let L be list drawn from the set X , then we define the dropoperator on L such that for any n 2 N , dropðn;LÞ is a list on X given by the

position function

Pdropðn;LÞðxÞ ¼u; if n > min PLðxÞ;fr � n : r6 n ^ r 2 PLðxÞg; if n6 min PLðxÞ

�

for any x 2 X .

Definition 2.10. For any two lists L1 and L2 defined over X and Y respectively,

we define their cartesian product L1 � L2 by

PL1�L2ðx; yÞ ¼ f#L2:sþ t : s 2 PL1ðxÞ; t 2 PL2ðyÞg:

Definition 2.11. Let L be a list and ½x� be a singleton list. Then the position

function of the list L� ½x� is defined by

PðL�½x�ÞðyÞ ¼fr : r 2 PLðyÞ and r < min PLðxÞg[fr � 1 : r 2 PLðyÞ and r > min PLðxÞg; for y 6¼ x;

fr � 1 : r 2 PLðxÞ and r > min PLðxÞg; for y ¼ x:

8<:

Definition 2.12. For any two lists L and L0, the list L� L0 is given by its

position function, defined recursively by

PL�L0 ðxÞ ¼PLðxÞ; if L0 is an empty list;PfL�½hdðL0Þ�g�tlðL0ÞðxÞ; otherwise:

�

Definition 2.13. Any two lists L and L0 drawn from X are said to be equal if and

only if

PLðxÞ ¼ PL0 ðxÞ; for all x 2 X :

Definition 2.14. Let L1 and L2 be two lists drawn from X . Then we definezipðL1;L2Þ as a list drawn from XxX and is given by its position function

PzipðL1;L2Þðx; yÞ ¼ PL1ðX Þ \ PL2ðY Þ:

170 B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179

3. Map operator on lists

In this section we introduce the notion of map operator on lists and study its

properties.

Definition 3.1. Let L be a list drawn from X . Let f be a mapping

defined from X to X . Then mapðf ; LÞ is a list whose position function is defined

by

Pmapðf ;LÞðyÞ ¼ fr : y ¼ f ðxÞ for some x 2 X ^ r 2 PLðxÞg:

It may be noted that in the above definition f : X ! X is considered to be an

onto map. However, in case f is not an onto map, map ðf ; LÞ will be a list

defined over f ðX Þ, where f ðX Þ denotes the range of f . Also, the definition is

applicable when f is either one-one or many-one.

For example, let X ¼ Z, L ¼ ½1;�1; 1; 2;�2;�2; 3� and f ðxÞ ¼ x2. Then map

ðf ; LÞ ¼ ½1; 1; 1; 4; 4; 4; 9�.Note: Since each element x in L is simply replaced by f ðxÞ in mapðf ; LÞ, it

can be noted that #L ¼ #mapðf ; LÞ.We require the following lemma in proving the theorems of this section.

Lemma. For any finite list L drawn from X , hdðLÞ ¼ lastðrevðLÞÞ.

Proof. Let y ¼ lastðrevðLÞÞ. Then, 0 2 PLðxÞ and #revðLÞ � 1 2 PrevðLÞðyÞ.So, by definition of revðLÞ,

y ¼ lastðrevðLÞÞ() #revðLÞ � 1 2 PrevðLÞðyÞ() ð#revðLÞÞ � ð#revðLÞ � 1Þ � 1 2 PLðyÞ.() 0 2 PLðyÞ() y ¼ hdðLÞ

Hence, hdðLÞ ¼ lastðrevðLÞÞ. �

Theorem 3.1. For any two functions f and g defined on X , and lists L, L1, and L2

drawn from X , where o denotes the usual composition of functions,

(i) mapðfog; LÞ ¼ mapðf ;mapðg; LÞÞ. Further, if the lists L, L1 and L2 arefinite, then

(ii) mapðf ; L1,L2Þ ¼ mapðf ; L1Þ,mapðf ; L2Þ(iii) mapðf ; revðLÞÞ ¼ revðmapðf ; LÞÞ(iv) hdðmapðf ; LÞÞ ¼ lastðmapðf ; revðLÞÞÞ(v) mapðf ; ½ �Þ ¼ ½ �.

B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179 171

Proof

(i) Since f , g: X ! X , it follows that fog: X ! X . For any y 2 fog ðX Þ,

Pmapðfog;LÞðyÞ ¼ fr : y ¼ ðfogÞðxÞ ^ r 2 PLðxÞg

¼ fr : y ¼ f ðgðxÞÞ ^ r 2 PLðxÞg

¼ fr : y ¼ f ðzÞ ^ z ¼ gðxÞ ^ r 2 PLðxÞg

¼ fr : y ¼ f ðzÞ ^ fr : z ¼ gðxÞ ^ r 2 PLðxÞgg

¼ fr : y ¼ f ðzÞ ^ r 2 Pmapðg;LÞðzÞg

¼ Pmapðf ;mapðg;LÞÞðyÞ:

(ii) For any y 2 f ðX Þ,

Pmapðf ;L1,L2ÞðyÞ ¼ fr : y ¼ f ðxÞ ^ r 2 PL1,L2ðxÞg

¼ fr : y ¼ f ðxÞ ^ r 2 fPL1ðxÞ ^ f#L1 þ t : t 2 PL2ðxÞggg¼ fr : ðy ¼ f ðxÞ ^ r 2 PL1ðxÞÞ ^ ðy ¼ f ðxÞ

^ r 2 f#L1 þ t : t 2 PL2ðxÞgÞg:

(Since PL1ðxÞ and f#L1 þ t : t 2 PL2ðxÞg are disjoint.)

¼ fr : ðy ¼ f ðxÞ ^ r 2 PL1ðxÞÞ ^ ðy ¼ f ðxÞ^ r 2 f#mapðf ; L1Þ þ t : t 2 fPL2ðxÞgÞg

¼ fr : ðy ¼ f ðxÞ ^ r 2 PL1ðxÞÞ ^ ðy ¼ f ðxÞ^ r 2 f#mapðf ; L1Þ þ t : t 2 Pmapðf ;L2 ÞðyÞgÞg

¼ Pmapðf ;L1Þ,mapðf ;L2ÞðyÞ:

(iii) For any y 2 f ðX Þ,

Pmapðf ;revðLÞÞðyÞ ¼ fr : y ¼ f ðxÞ ^ r 2 PrevðLÞðxÞg

¼ fr : y ¼ f ðxÞ ^ r 2 f#L� t � 1 : t 2 PLðxÞgg¼ fr : y ¼ f ðxÞ ^ r 2 f#mapðf ;LÞ � t � 1 : t 2 Pmapðf ;LÞðxÞgg¼ f#mapðf ;LÞ � t � 1 : y ¼ f ðxÞ ^ t 2 Pmapðf ;LÞðxÞg¼ Prevðmapðf ;LÞÞðyÞ:

(iv) From part (iii) above and using Lemma,

hdðmapðf ; LÞÞ ¼ lastðrevðmapðf ; LÞÞÞ ¼ lastðmapðf ; revðLÞÞÞ:

(v) For any y 2 f ðX Þ,

Pmapðf ;½ �ÞðyÞ ¼ fr : y ¼ f ðxÞ ^ r 2 P½ �ðxÞg ¼ fr : y ¼ f ðxÞ ^ r 2 ug¼ u ¼ P½ �ðyÞ:

172 B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179

Theorem 3.2. For any list L drawn from X , f being a function on X and n 2 N ,

(i) mapðf ; takeðn; LÞÞ ¼ takeðn;mapðf ; LÞÞ(ii) mapðf ; dropðn; LÞÞ ¼ dropðn;mapðf ; LÞÞ.

Proof. For any n 2 N ,

(i) Pmapðf ;takeðn;LÞÞðyÞ ¼ fr : y ¼ f ðxÞ ^ r 2 Ptakeðn;LÞðxÞg¼ fr : y ¼ f ðxÞ ^ r 2 ft : t < n ^ t 2 PLðxÞgg¼ fr : y ¼ f ðxÞ ^ r < n ^ r 2 PLðxÞg¼ fr : r < nfr : y ¼ f ðxÞ ^ r 2 PLðxÞgg¼ fr : r < n ^ r 2 Pmapðf ;LÞðyÞg¼ Ptakeðn;mapðf ;LÞÞðyÞ.

(ii) Pmapðf ;dropðn;LÞÞðyÞ ¼ fr : y ¼ f ðxÞ ^ r 2 Pdropðn;LÞðxÞg¼ fr : y ¼ f ðxÞ ^ r 2 ft � n : tP n ^ t 2 PLðxÞgg¼ fr : y ¼ f ðxÞ ^ rP n ^ r 2 PLðxÞg¼ fr : rP n ^ fr : y ¼ f ðxÞ ^ r 2 PLðxÞgg¼ fr : rP n ^ r 2 Pmapðf ;LÞðyÞg¼ Pdropðn;mapðf ;LÞÞðyÞ.

Theorem 3.3. For any function f defined on X , and two lists L1, L2 drawn from X ,

(i) mapðf ; zipðL1; L2ÞÞ ¼ zipðmapðf ; L1Þ;mapðf ; L2ÞÞ(ii) mapðf ; L1 � L2Þ ¼ mapðf ; L1Þ �mapðf ; L2Þ.

Proof

(i) Pzipðmapðf ;L1Þ;mapðf ;L2ÞÞðy1; y2Þ ¼ Pmapðf ;L1Þðy1Þ \ Pmapðf ;L2Þðy2Þ¼ fr : y1 ¼ f ðx1Þ ^ r 2 PL1ðx1Þg \ ft : y2 ¼ f ðx2Þ ^ t 2 PL2ðx2Þg¼ fs : ðy1 ¼ f ðx1Þ ^ y2 ¼ f ðx2ÞÞ ^ s 2 ðPL1ðx1Þ \ PL2ðx2ÞÞg¼ fs : ðy1 ¼ f ðx1Þ ^ y2 ¼ f ðx2ÞÞ ^ s 2 PzipðL1;L2Þðx1; x2Þg¼ Pmapðf ;zipðL1;L2Þðy1; y2Þ.

(ii) Pmapðf ;L1�L2Þðy1; y2Þ ¼ fr : y1 ¼ f ðx1Þ ^ y2 ¼ f ðx2Þ ^ r 2 PL1�L2ðx1; x2Þg¼ fr : y1 ¼ f ðx1Þ ^ y2 ¼ f ðx2Þ ^ r 2 f#L2:sþ t : s 2 PL1ðx1Þ; t 2 PL2ðx2Þg¼ f#L2:sþ t : ðy1 ¼ f ðx1Þ ^ s 2 PL1ðx1ÞÞ ^ ðy2 ¼ f ðx2Þ ^ t 2 PL2ðx2ÞÞg¼ f#mapðf ; L1Þ:sþ t : s 2 Pmapðf ;L1Þðy1Þ ^ t 2 Pmapðf ;L2Þðy2Þg¼ Pmapðf ;L1Þ�mapðf ;L1Þðy1; y2Þ.

For any two lists L and L0 and a map f from X into itself, it may not be truethat mapðf ; L� L0Þ ¼ mapðf ; LÞ �mapðf ; L0Þ.

Even the particular case that for any x 2 X , mapðf ; L� ½x�Þ ¼ mapðf ; LÞ�mapðf ; ½x�Þ, may not be true.

To establish this we provide an example. Let L ¼ ½1;�1; 2;�2; 1; 3;�1;�3; 2�and f : x 2 X be defined as f ðxÞ ¼ x2. Taking x ¼ �3; L� ½�3� ¼ ½1;�1; 2;�2;1; 3;�1; 2�, mapðf ; L� ½�3�Þ ¼ ½1; 1; 4; 4; 1; 9; 1; 4� and mapðf ; ½�3�Þ ¼ ½9�. So

B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179 173

mapðf ;LÞ �mapðf ; ½�3�Þ ¼ ½1;1;4;4;1;1;9;4�.Hence mapðf ; L� ½�3�Þ 6¼ mapðf ; LÞ �mapðf ; ½�3�Þ.

Theorem 3.4. For any list L and a finite list L0 drawn from X , a one to onemapping ‘f ’ from X to X and x 2 X ,

(i) mapðf ; L� ½x�Þ ¼ mapðf ; LÞ �mapðf ; ½x�Þ(ii) mapðf ; L� L0Þ ¼ mapðf ; LÞ �mapðf ; L0Þ:

Proof

(i) Let f ðxÞ ¼ w. Then for any y 2 f ðX Þ,

Pmapðf ;L�½x�ÞðyÞ ¼nr : y ¼ f ðzÞ ^ r 2 P ðzÞ

L�½x�

o

¼ fr : y ¼ f ðzÞ ^ r 2 ft : t 2 PLðzÞg ^ t < min PLðxÞg[ fu� 1 : u 2 PLðzÞ ^ u > min PLðxÞÞg

¼ fr : y ¼ f ðzÞ ^ r 2 PLðzÞ ^ r < min PLðxÞg[ fr � 1 : y ¼ f ðzÞ ^ r 2 PLðzÞ ^ r > min PLðxÞg

¼ fr : r 2 Pmapðf ;LÞðyÞ ^ r < min Pmapðf ;LÞðwÞg[ fr � 1 : r 2 Pmapðf ;LÞðyÞ ^ r > min Pmapðf ;LÞðwÞg

¼ Pmapðf ;LÞ�mapðf ;½x�ÞðyÞ:

This completes the proof of (a).

Now, for any L0, L0 ¼ ½hdðL0Þ�,tlðL0Þ. So,

L� L0 ¼ L� ½hdðL0Þ� � tlðL0Þ

¼ ððL� ½hdðL0Þ�Þ � ½hdðtlðL0ÞÞ� � � � � � ½hdðtlðtlð- - - -ðtlðL0ÞÞ . . .Þ�:

Applying part (a) successively, we get (b).

4. A property of ‘take’ and ‘drop’ operators

In this section we prove the following theorem on the ‘take’ and ‘drop’

operator on a list.

Theorem. If L1 and L2 are the lists drawn from X and f is a mapping from X intoX , then for any n 2 N ,

ðaÞ takeðn; L1,L2Þ ¼ takeðn; L1Þ; if n6 #L1;¼ L1,takeðn� 6¼ L1; L2Þ; if n > #L1:

174 B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179

ðbÞ dropðn; L1,L2Þ ¼ dropðn; L1Þ,L2; if n6 #L1;¼ dropðn� ,L1; L2Þ; if n > #L1:

Proof. (a) Case (i) Suppose n6#L1. Then, for any x 2 X ,

Ptakeðn;L1,L2ÞðxÞ ¼ fr : r < n ^ r 2 PL1,L2ðxÞg¼ fr : r < n ^ r 2 PL1ðxÞ [ f#L1þt : t 2 PL2ðxÞgg:

ð4:1Þ

But, here n6#L1. So,

r < n ) r < #L1 ) r 62 f#L1þ t : t 2 PL2ðxÞg:

Hence,

LHS ofð4:1Þ ¼ fr : r < n ^ r 2 PL1ðxÞg¼ Ptakeðn;L1ÞðxÞ:

So, takeðn; L1,L2Þ ¼ takeðn; L1Þ.Case (ii) Suppose n > #L1. Then, takeðn; L1Þ ¼ L1. So, from (4.1),

Ptakeðn;L1,L2ÞðxÞ ¼ fr : r< n^ r 2 PL1ðxÞg[ fr : r< n^ r 2 f#L1 þ t : t 2 PL2ðxÞg¼ fr : r 2 Ptakeðn;L1ÞðxÞg[ f#L1 þ t :#L1 þ t< n^ t 2 PL2ðxÞg¼ fr : r 2 Ptakeðn;L1ÞðxÞg[ f#L1 þ t : t< n�#L1 ^ t 2 PL2ðxÞg¼ fr : r 2 PL1ðxÞg[ f#L1 þ t : t 2 Ptakeðn�#L1;L2ÞðxÞg¼ fr : r 2 PL1,takeðn�#L1;L2ÞðxÞg:

So, takeðn; L1,L2Þ ¼ L1,takeðn� #L1; L2Þ.(b) Two cases arise here.

Case (i) n > max PL1,L2ðxÞ. Again we consider two cases under it.

Case (i) (1) Suppose n6#L1. Then x 62 L2. For, otherwise

max PL1,L2ðxÞ > #L1 P n

which is a contradiction. So, PL2ðxÞ ¼ /. Hence,

Pdropðn;L1Þ,L2ðxÞ ¼ fr : r 2 Pdropðn;L1ÞðxÞg [ f#dropðn; L1Þ þ t : t 2 PL2ðxÞg¼ fr : r 2 Pdropðn;L1ÞðxÞg:

ð4:2ÞBut; n > max PL1,L2ðxÞ ) n > max PL1ðxÞ

) Pdropðn;L1ÞðxÞ ¼ /:

So, LHS of (4.2)¼/. Also, n > max PL1,L2 ðxÞ implies

Pdropðn;L1,L2ÞðxÞ ¼ /:

Hence, Pdropðn;L1,L2ÞðxÞ ¼ Pdropðn;L1Þ,L2ðxÞ.

B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179 175

Case (i) (2) Suppose n > #L1. Then dropðn; L1Þ ¼ /. So,

Pdropðn�#L1;L2ÞðxÞ ¼ fr : rP n� #L1 ^ r 2 PL2ðxÞg

¼ fr : #L1 þ rP n ^ r 2 PL2ðxÞg

¼ /

as n > max PL1,L2ðxÞ ) n > #L1 þ r for all r 2 PL2ðxÞ.

Hence, Pdropðn;L1#L2ÞðxÞ ¼ Pdropðn�#L1;L2ÞðxÞ.Case (ii) Suppose n6 max PL1,L2ðxÞ. Two cases arise.

Case (ii) (1) Let n < #L1. Then n < #L1 þ #L2. So that

Pdropðn;L1,L2ÞðxÞ ¼ fr � n : rP n ^ r 2 PL1,L2ðxÞg

¼ fr � n : rP n ^ r 2 PL1ðxÞ [ f#L1 þ t : t 2 PL2ðxÞgg

¼ fr � n : rP n ^ r 2 PL1ðxÞ

[ fr � n : rP n ^ r 2 f#L1 þ t : t 2 PL2ðxÞgg

¼ fr � n : rP n ^ r 2 PL1ðxÞ

[ f#L1 þ t � n : #L1 þ tP n ^ t 2 PL2ðxÞg

¼ fr � n : rP n ^ r 2 PL1ðxÞ

[ f#dropðn; L1Þ þ t : t 2 PL2ðxÞg

¼ Pdropðn;L1ÞðxÞ [ f#dropðn; L1Þ þ t : t 2 PL2ðxÞg

¼ Pdropðn;L1Þ,L2ðxÞ:ð4:3Þ

Case (ii) (2) Let nP#L1. Then from (4.3)

Pdropðn;L1,L2ÞðxÞ ¼ fr � n : rP n ^ r 2 PL1ðxÞg

[ fr � n : rP n ^ r 2 f#L1 þ t : t 2 P ðxÞgg

¼ / [ fr � n : rP n ^ r 2 f#L1 þ t : t 2 PL2ðxÞgg

¼ f#L1 þ t � n : #L1 þ tP n ^ t 2 PL2ðxÞg

¼ ft � ðn� #L1Þ : tP n� #L1 ^ t 2 PL2ðxÞg

¼ Pdropðn�#L1;L2ÞðxÞ:

This completes the proof. �

176 B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179

5. Some other operators on lists

In this section we introduce the notions list indexing, cons operator and

prove some results basing upon them, which provide some applications of themap operator besides being results of individual interest.

Definition 5.1. For any list L drawn from X and any natural number n, wedefine the element indexðL; nÞ as x 2 X , such that x ¼ hdðtakeðnþ 1; LÞ�takeðn; LÞÞ.

Definition 5.2. For any x 2 X and a list L drawn from X , we denote the list

which is obtained by adding an element x at the beginning of the list L byconsðx;LÞ and define it by its membership function,

Pconsðx;LÞðyÞ ¼fr þ 1 : r 2 PLðyÞg; if y 6¼ x;

f0g [ fr þ 1 : r 2 PLðxÞg; if y ¼ x:

�

We now establish some properties of the notions introduced in this section.

Lemma. For any three lists L,M and N drawn from X , we have

ðL,MÞ � ðL,NÞ ¼ M � N :

Proof. We have,

ðL,MÞ � ðL,NÞ ¼ ðL,M � ½hdðL,NÞ� � tlðL,NÞ

¼ ðL,MÞ � ½hdðLÞ� � tlðL,NÞ

¼ tlðL,MÞ � tlðL,NÞ:

ð5:1Þ

Proceeding like this ð#L� 1Þ times, we get

LHS ofð5:1Þ ¼ tlð½lastðLÞ�,MÞ � tlð½lastðLÞ�,NÞ

¼ M � N :

This completes the proof. �

Theorem 5.1. For any n 2 N and for lists L1 and L2 drawn from X ,

indexðL1#L2; nÞ ¼index ðL1; nÞ; if n < #L1;

index ðL2; n� #L1Þ; if n#L1:

�

B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179 177

Proof. If n < #L1 then by the theorem of Section 4 we have

indexðL1,L2; nÞ ¼ hdðtakeðnþ 1; L1,L2Þ � takeðn; L1,L2ÞÞ¼ hdðtakeðnþ 1; L1Þ � takeðn; L1ÞÞ¼ indexðL1; nÞ:

If n ¼ #L1,

indexðL1,L2; nÞ ¼ hdðL1,½hdðL2Þ� � L1Þ¼ hdð½hdðL2Þ�Þ¼ hdðL2Þ¼ indexðL2; 0Þ � indexðL2; n� #L1Þ:

If n > #L1 then takeðn; L1Þ ¼ L1. So, by the lemma of this section,

indexðL1,L2; nÞ ¼ hdðtakeðnþ 1; L1,L2Þ � takeðn; L1,L2ÞÞ¼ hdðL1,takeðnþ 1� #L1; L2Þ � ðL1,takeðn� L1; L2ÞÞ¼ hdðtakeðnþ 1� #L1; L2Þ � takeðn� #L1; L2ÞÞ¼ indexðL2; n� #L1Þ:

This completes the proof. �

Theorem 5.2. For any x 2 X and a list L drawn from X ,

(i) hdðconsðx; LÞÞ ¼ x(ii) tlðconsðx; LÞÞ ¼ L(iii) reverse ðL,½x�Þ ¼ consðx; reverseðLÞÞ:

Proof

(i) If y ¼ hdðconsðx; LÞÞ then0 2 Pconsðx;LÞðyÞ. So, y ¼ x.

(ii) If y 6¼ x then

Ptailðconsðx;LÞÞðyÞ ¼ fr � 1 : r 2 Pconsðx;LÞðyÞ ^ rP 1g¼ fr � 1 : r ¼ t þ 1 ^ t 2 PLðyÞ ^ rP 1g¼ ft : t 2 PLðyÞ ^ tP 0g¼ PLðyÞ:

Ptailðconsðx;LÞÞðxÞ ¼ fr � 1 : r 2 Pconsðx;LÞðxÞ ^ rP 1g¼ fr � 1 : r 2 f0g [ ft þ 1 : t 2 PLðxÞ ^ rP 1gg¼ fr � 1 : r 2 ft þ 1 : t 2 PLðxÞ ^ rP 1gg¼ ft : t 2 PLðxÞg¼ PLðxÞ:

Hence, tail(consðx; LÞ)¼ L.

178 B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179

(iii) If y 6¼ x,

PrevðL,½x�ÞðyÞ ¼ f#ðL� ½x�Þ � t � 1 : t 2 PL,½x�ðyÞg

¼ f#L� t : t 2 PL,½x�ðyÞg

¼ f#L� t : t 2 PLðyÞ [ f#Lþ u : u 2 P½x�ðyÞgg

¼ f#L� t : t 2 PLgðyÞ:

ð5:2Þ

Pconsðx;revðLÞÞðyÞ ¼ fr þ 1 : r 2 PrevðLÞðyÞg

¼ fr þ 1 : r 2 f#L� u� 1 : u 2 PLðyÞgg

¼ f#L� u : u 2 PLðyÞg:

ð5:3Þ

From (5.2) and (5.3), we have

PrevðL,½x�ÞðyÞ ¼ Pconsðx;revðLÞÞðyÞ

PrevðL,½x�ÞðxÞ ¼ f#L� t : t 2 PLðxÞ [ f#Lþ u : u 2 P½x�ðxÞgg

¼ f#L� t : t 2 PLðxÞyf#Lgg

¼ f0gyf#L� t : t 2 PLðxÞg

¼ f0g [ fr þ 1 : r 2 PrevðLÞðxÞg

¼ Pconsðx;revðLÞÞðxÞ:

Hence, revðL,½x�Þ ¼ consðx; revðLÞÞ. h

6. Fuzzy lists

In this section, we introduce the fuzzy version of the notions introduced

above.

Definition 6.1. Let L be a fuzzy list [3] drawn from X . Let f be a mapping

defined from X to X . Then map ðf ; LÞ is a fuzzy list, whose position functionis given by

Pmapðf ;LÞðy; aÞ ¼ fr : r ¼ f ðxÞ for some x 2 X ^ r 2 PLðx; aÞg:

Here also f may be considered as an onto map. However, is case f is not an

onto map, mapðf ; LÞ will be a fuzzy list defined over f ðX Þ.

Definition 6.2. For any fuzzy list L drawn from X and any natural number n,we define the element index (L, n) as ðx; aÞ 2 L such that

ðx; aÞ ¼ hdðtakeðnþ 1; LÞ � takeðn; LÞÞ:

B.K. Tripathy, S.S. Gantayat / Information Sciences 166 (2004) 167–179 179

Definition 6.3. For any x 2 X ; a 2 I and a fuzzy list L drawn from X , we denote

the fuzzy list, which is obtained by adding an element ðx; aÞ at the beginning of

the fuzzy list L by cons ððx; aÞ; LÞ and define it by its membership function,

Pðcansðx;aÞ;LÞðy; bÞ ¼fr þ 1 : r 2 PLðy; bÞg; if y 6¼ x;f0g [ fr þ 1 : r 2 PLðx; bÞg; if y ¼ x:

�

It may be noted that all the results established above for lists can be extended

to fuzzy lists without any difficulty.

7. Conclusion

In this article we have established another property of the operators ‘take’

and ‘drop’ on a list [4,5]. Also, we defined the important notion of ‘map’ on a

list and the operators ‘cons’ and ‘index’ on it. We have proved several results,

which establish the properties of these notions. These notions seem to have

wide applications, which need to be explored. The parallel notions for fuzzy

lists have been introduced and studied.

References

[1] R. Bird, P. Walder, Introduction to Functional Programming, Prentice Hall, Englewood Cliffs,

NJ, 1988.

[2] S.P. Jena, S.K. Ghosh, B.K. Tripathy, On the theory of bags and lists, Information Sciences 132

(1–4) (2001) 241–254.

[3] S.P. Jena, S.K. Ghosh, B.K. Tripathy, On the theory of fuzzy bags and fuzzy lists, The Journal

of Fuzzy Maths 9 (4) (2001) 1209–1220.

[4] B.K. Tripathy, G.P. Patnaik, On some properties of lists and fuzzy lists, in: Proceedings of the

Conference on Fuzzy set Theory and its Mathematical Aspects and Applications, Banaras

Hindu University, Banaras, December 2002, pp. 234–238.

[5] B.K. Tripathy, G.P. Patnaik, On some properties of lists and fuzzy lists, Information Sciences,

in press.

[6] R.R. Yager, On the theory of bags, International Journal of General Systems 13 (1986) 23–37.