set operations pdf

2. Let us discuss the important operations here: The important operations on sets are. operations and that is not too large to be moved from one work site to another. $O./� �'�z8�W�Gб� x�� 0Y驾A��@$/7z�� ���H��e��O���OҬT� �_��lN:K��"N����3"��$�F��/JP�rb�[䥟}�Q��d[��S��l1��x{��#b�G�\N��o�X3I���[ql2�� �$�8�x����t�r p��/8�p��C���f�q��.K�njm͠{r2�8��?�����. Sets and Set Operations Class Note 04: Sets and Set Operations Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 45 Sets Denition: ASetis acollection of objectsthat do NOT have an order. )ɩL^6 �g�,qm�"[�Z[Z��~Q����7%��"� Set operations in LINQ refer to query operations that produce a result set that is based on the presence or absence of equivalent elements within the same or separate collections (or sets). (The common element occurs only once) You can change your ad preferences anytime. Set operations in LINQ refer to query operations that produce a result set that is based on the presence or absence of equivalent elements within the same or separate collections (or sets). … These are called op-erator precedence rules. A trained operator can accomplish more machining jobs with the engine lathe than with any other machine tool. In a set theory, there are three major types of operations performed on sets, such as: Union of sets (∪) Intersection of sets (∩) Difference of sets ( – ) Let us discuss these operations one by one. 83 0 obj <>/Filter/FlateDecode/ID[<7699FE2A76498BA3504AB9257FEAFED9>]/Index[77 17]/Info 76 0 R/Length 53/Prev 67195/Root 78 0 R/Size 94/Type/XRef/W[1 2 1]>>stream • N = {1, 2, 3, ... } • The set of reals is an infinite set. Complement 6. 26 CHAPTER 2. �tq�X)I)B>==���� �ȉ��9. set operations. B belongs to both A and B, an element of A # B is required to belong to at least one of the sets. In addition to this operator notation, there are method functions which do the same things. set in the family a "label" called an index, which need not be related in any way to the elements of the set. (ii) Operations between parenthesis are done first, �u�Q��y�V��|�_�G� ]x�P? 2.2 Set Operations Union The union of the sets A and B, denoted by A [B, is the set that contains those elements that are either in A or in B, or in both. Set Operations Complement: The complement of a set A is the set of all elements in the universal set NOT contained in A, denoted A. A = {Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia} Set B below contains the five best films according to TV Guide. BASIC SET THEORY (i) Other things being equal, operations are per-formed left-to-right. of set theory were a real threat to the security of the foundations. (Cantor's naive definition) • Examples: – Vowels in the English alphabet V = … Set Operations Operations between sets allow us to examine and manipulate the contents of sets in ways similar to logical and Boolean operations. Statement (2) is true; it is called the Schroder-Bernstein hޜ�wTT��Ͻwz��0�z�.0��. The set of all indices, often denoted by ∆ is called an indexing set. In fuzzy logic, three operations, including fuzzy complement, fuzzy intersection and fuzzy union, are the most commonly used. 0 Statement (2) is true; it is called the Schroder-Bernstein 336 0 obj <> endobj xref 336 14 0000000016 00000 n 4 Whitehead’s theory of strati ed types and then more elegantly, in for exam-ple the in uential work of Zermelo and Fraenkel. The set of all indices, often denoted by ∆ is called an indexing set. (i) Commutative Property : (a) A u B = B u A (Set union is commutative) (b) A n B = B n A (Set … (Caution: sometimes ⊂ is used the way we are using ⊆.) 0000002075 00000 n The standard query operator methods that perform set operations are listed in the following section. 0000001713 00000 n E. and . 0000001598 00000 n A is the set of multiples of 3. 0000000576 00000 n ��8SJ?����M�� ��Y ��)�Q�h��>M���WU%qK�K0$�~�3e��f�G�� =��Td�C�J�b�Ҁ)VHP�C.-�7S-�01�O7����ת��L:P� �%�",5�P��;0��,Ÿ0� 2.2 Set Operations 1. A[B = fx jx 2A_x 2Bg Intersection The intersection of the sets A and B, denoted by A \B, is the set containing those elements in both A … 0000001221 00000 n Sometimes the complement is denoted as A' or AC. %PDF-1.4 %���� x�b```a``� Value A list with three named components: set The set created from x. mappingmapping, possibly reordered to match the order of set. The difference between sets is denoted by ‘A – B’, which is the set containing elements that are in A but not in B. complement of set ordered pair, ordered n-tuple equality of ordered n-tuples Cartesian product of sets Contents Sets can be combined in a number of different ways to produce another set. 0000005436 00000 n A = { Mary, Mark, Fred, Angela, Frank, Laura } B = { Fred, Mary, Frank, Jane } SET OPERATIONS, VENN DIAGRAMS SET OPERATIONS Let U = {x|x is an English-language film} Set A below contains the five best films according to the American Film Institute. trailer <<488D8812050A4AB8B4AAC4DB5D9E1639>]>> startxref 0 %%EOF 349 0 obj <>stream Since we're doing the same manipulations, we ended up with the same tables. 2 Union Let A and B be sets. Subsets A set A is a subset of a set B iff every element of A is also an element of B.Such a relation between sets is denoted by A ⊆ B.If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. 0000002743 00000 n 0000001635 00000 n Methods. Union 2. There are a large number of set operations, including union (|), intersection (&), difference (-), symmetric difference (^). They won’t appear on an assignment, however, because they are quite dif-7. Example Of UNION Table A Table B UNION Set Operator SQL Query SQL> SELECT * FROM A UNION SELECT * FROM B Result of the above UNION Operator will be An Introduction To Sets, Set Operations and Venn Diagrams, basic ways of describing sets, use of set notation, finite sets, infinite sets, empty sets, subsets, universal sets, complement of a set, basic set operations including intersection and union of sets, and applications of sets, with video lessons, examples and step-by-step solutions. Here are some useful rules and definitions for working with sets Disjoint sets Let us discuss the above operations in detail one by one. Subsets A set A is a subset of a set B iff every element of A is also an element of B.Such a relation between sets is denoted by A ⊆ B.If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. By the use of this function, the meta information can be kept in sync with the result of iterating over the associated set. E. be the set of days in June. Set Operations 1. hޤV[o�0�+�q{`���H��UZ;Ԡu�! 0000002389 00000 n Program should check the provided input to check whether its valid or not. CS 441 Discrete mathematics for CS M. Hauskrecht Set • Definition: A set is a (unordered) collection of objects. The following are the important properties of set operations. A # B = { x | x " A or x " B } This is the union of A and B. These are called op-erator precedence rules. Set Operations. Sometimes the complement is denoted as A' or AC. endstream endobj 81 0 obj <>stream Set Operations • The union of two sets Aand B, written A∪ B, is the set of all elements that are IN AOR B OR BOTH. &.��M,[email protected]���#�,"I,��*�]�: INTRODUCTION ficult to prove. Sets and set operations: cont. (Caution: sometimes ⊂ is used the way we are using ⊆.) endstream endobj 343 0 obj [/Pattern 340 0 R] endobj 344 0 obj <>stream Union of Sets. 9 CS 441 Discrete mathematics for CS M. Hauskrecht Power set Definition: Given a set S, the power set of S is the set of all subsets of S. endstream endobj 345 0 obj <> endobj 346 0 obj <>stream "�Wk��αs�[[d�>7�����* !BP!����P�K*�8 �� ��..ؤȋ29�+MJR:��!�z2׉I 9�A�cZ� ��sIeІ�O5�Rz9+�U�͂�.�l���r8\���d�Vz ��-1���N�J�p�%�ZMn��͟�k����Z��Q����:�l �9���5�"d�|���#�MW���N�]�?�g;]�����.����t������g��ܺSj�ڲ��ܥ�5=�n|l�Ƥy��7���w?��dJ͖��%��ŽH�E1/�گ�u�߰�l?�WY�O��2�mZ�'O Union: The union of two sets is the set of elements that belong to either of the two h��UM��6��W�Q* �_"��8�A}h-��E^[^k㵼��m~H�{3CR�� ����L��p�7�O����Z �5���@W'�DŽ�-%� i.e., all elements of A except the element of B. Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 4 Set Theory Basics.doc 1.4. Worksheet 2 Sets – Set Operations 1. %PDF-1.5 %���� A set can be represented by listing its elements between braces: A = {1,2,3,4,5}.The symbol ∈ is used to express that an element is (or belongs to) a set… The standard query operator methods that perform set operations are listed in the following section. We could introduce … 3�+\! h�bbd``b`�$�C�`���@�+#��#1�Ɗ *� Intersection 3. Programs source code to perform Set Operations program in C and C++ Language.Operations on set to be performed are Given below. Example: Let A = {1, 3, 5, 7, 9} and B = { 2, 4, 6, 8} A and B are disjoint sets since both of them have no common elements. U is the set of whole numbers from 1 to 15. D i s c re teS tru c tu re s (Discrete Mathematics) Topic: Set Operations ©bilalAmjad [email protected] h�b```f``�d`b``Kg�[email protected] ^�3�Cr��N?_cN� � W���&����vn���W�}5���>�����������l��(���b E�l �B���f`x��Y���^F��^��cJ������4#w����Ϩ` <4� R. be the set of rainy days. ���@��~���˲���T�Y�쟗�1r��B5WG��#-�3�� f�{����v��7�r��uT����M�X&vF�O(�ΥĔ���#b�P���^]����ܵ�Uw� hLB � 3�Nn���)�q=f�.�_M�E�Q6m�&�MT� �?t ex) U={integers from 1 to 10} A={3,6,9}, A={1,2,4,5,7,8,10} which are all elements from the universal set that are not found in A. 77 0 obj <> endobj %%EOF 0000005472 00000 n When two or more sets are combined together to form another set under some given conditions, then operations on sets are carried out. (ii) Operations between parenthesis are done first, Set Operations Complement: The complement of a set A is the set of all elements in the universal set NOT contained in A, denoted A. K��hThj�)x��ɑ�M��#�#��B'C���*5�V]���#��;s�l�l��뢗��}� �x�).C��R*�@�M:�6��,j9)s�2�aW���]y6sU(�Z}cm��GǶ�yO/�M� ����Č�[email protected]��� * P��� D��� B(�R2����� �P�+� F�i [email protected]���ѣ��(�/�;�47ǃETx�1h�$0�+�-``O�c��ɷ�WL ��B�؆, X|�.��m��J��2��\�f�f����1���C3Q?�?���,�7ƱS��!�dK>Lbyp��a�h��D����b ���CT!H|�oC������’[email protected]� ��3��I �;� V��� endstream endobj 337 0 obj <> endobj 338 0 obj <> endobj 339 0 obj <>/Font<>/ProcSet[/PDF/Text]/ExtGState<>/Pattern<>>> endobj 340 0 obj [/ICCBased 346 0 R] endobj 341 0 obj <> endobj 342 0 obj <>stream (�dg)*�+(�*D�(�[email protected]�A����Br.��֙��$m�!�� h Above is the Venn Diagram of A disjoint B. Turret lathes and special purpose lathes are usually used in production or job shops for endstream endobj 78 0 obj <> endobj 79 0 obj <> endobj 80 0 obj <>stream These objects are sometimes called elements or members of the set. Operations on sets : When two or more sets are combined together to form another set under some given conditions, then operations on sets are carried out. B = { x | x " A and x " B } This is the intersection of A and B. Example− If A = { 10, 11, 12, 13 } and B = { 13, 14, 15 }, then A ∪ B = { 10, 11, 12, 13, 14, 15 }. Set operations can be used to combine sets. 26 CHAPTER 2. Create a Venn diagram to show the relationship among the sets. h�*�2T�T�2P0P� ¢T. We Fuzzy set operations are a generalization of crisp set operations, each of which is a fuzzy set operation. Set difference 4. Set Theory 2.1.1. Set operations and Venn diagrams A ! "��@ (�����.�'R�M�]L�x�����H�����$6W���\��@������4^3�e�b�R�o��r?�(T&���P1k��U�f��1��k9� Functions. The union of sets A and B (denoted by A ∪ B) is the set of elements that are in A, in B, or in both A and B. �M�,� S)���r����� Figure 1.2 Ac is shaded. Let fuzzy sets A and B be described by their membership functions μ A (x) and μ B (x).The three fuzzy set operations are defined below. ex) U={integers from 1 to 10} A={3,6,9}, A={1,2,4,5,7,8,10} which are all elements from the universal set that are not found in A. Just because it worked for these, doesn't mean you can assume everything is the same. Symmetric difference 5. Let U = {1,2, …, 9} be the universal set, and let A = We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. The union of A and B, denoted by A B, is the set containing those elements that are either in A or in B, or in both. 1) P is non-empty; 2) A∩B ∈ P whenever A, B ∈ P. Definition 0.0.7 (λ-system) Given a set Ω a λ system is a collection of subsets L that contains Ω and is closedunder complementation and disjoint countable unions. operations. We could introduce … Set Operations The first set operation we consider is the complement. But with a lot of worry and care the paradoxes were sidestepped, rst by Russell and. Sets and set operations ... • The set of natural numbers is an infinite set. W. and 3; together have size 25, so the overlap between W and R is 10.; The Venn diagram opposite displays the whole situation. A = {Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia} Set B below contains the five best films according to TV Guide. For the following examples, we will define two sets, A and B. Example: Consider the family F of half-open intervals of real numbers, [0,r). 0000001306 00000 n H�[}K�`G���2/�m��S�ͶZȀ>q����y��>`�@1��)#��o�K9)�G#��,zI�mk#¹�+�Ȋ9B*�!�|͍�6���-�I���v���f":��k:�ON��r��j�du�������6Ѳ��� �h�/{�%? C is the set of odd numbers 2. operations. The complement of set A are those members of set U that do not belong to A. Let . Hence, A ∪ B = { x | x ∈ A OR x ∈ B }. ����?���'�ف����˞y&�� In contrast, we provide efficient solutions for private multi-party Set-Intersection secure against malicious players, and our multiset intersection operator can be easily composed with other operations to enable a wide range of efficient private computation over multisets. Sets. Be careful with the other operations. Each object is called anelement. Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 4 Set Theory Basics.doc 1.4. h�t�MK1����Q�N'�4�^-"Ve�ò��~�n���n+X-��d�>��Fi�PƓ�p��bb�0��z�J���C�A������x�΅� H Then . Definition : The union of sets A and B, denoted by A B, is the set defined as *�1��'(�[P^#�����b�;_[ �:��(�JGh}=������]B���yT�[�PA��E��\���R���sa�ǘg*�M��cw���.�"M޻O��6����'Q`MY�0�Z:D{CtE�����)Jm3l9�>[�D���z-�Zn��l���������3R���ٽ�c̿ g\� B is the set of primes. Input Operations – This operation should allow the user to provide input to the program. be the set of windy days, W R. 5 10 10 5. Set Operations Niloufar Shafiei. ����,wi����f��C�>�g�I�$To1$W>6��x�/���2&R�����M$W����R1Ԁ1�)�p!#�L���ZL������p.=��|�f �����|Jm���`�r��KP΄��E�c����p�j��e֝�Y*�etf���H6/�C�#A��c�$cV�T�����8�u$�|�>feJ1��ѡ� ���ZZ�nzvj����sT��Izԥ�@��9T1�0�/���Z�$��Znb�~D�J�����v )��P��d��lT9s. A set is a collection of objects, called elements of the set. This set operator is used to combine the outputs of two or more queries into a single set of rows and columns having different records. 6 Definition 0.0.6 (π-system) Given a set Ω a π system is a collection of subsets P that are closed under finiteintersections. 1 Set operations Two sets can be combined in many different ways. There is a set of rules that reduces the number of parenthesis required. 2.2 Set Operations Union The union of the sets A and B, denoted by A [B, is the set that contains those elements that are either in A or in B, or in both. The engine lathe (Figure 7-1) is ideally suited for this purpose. 0000001448 00000 n For any one of the set operations, we can expand to set builder notation, and then use the logical equivalences to manipulate the conditions. Example: Consider the family F of half-open intervals of real numbers, [0,r). Set Operations and Venn Diagrams - Part 2 of 2 Examples: 1. CHAPTER 2 Sets, Functions, Relations 2.1. These are unusual operations, so we'll look at them in some detail. They won’t appear on an assignment, however, because they are quite dif-7. 8 CHAPTER 0. The purpose of this module is to introduce language for talking about sets, and some Complement Given a universal setU and a set A⊂U, the complement of A, written Ac, is the set of all elements that are in U but not in A, that is, Ac ={x|x∈U, x ∈/ A} Here four basic operations are introduced and their properties are discussed. SET OPERATIONS, VENN DIAGRAMS SET OPERATIONS Let U = {x|x is an English-language film} Set A below contains the five best films according to the American Film Institute. View Worksheet-2-Sets-Set-Operations (1).pdf from IST 230 at Pennsylvania State University, Abington. ��3�������R� `̊j��[�~ :� w���! INTRODUCTION ficult to prove. function from the set of real numbers into X or there is a one-to-one function from X into the set of rational numbers. 0000002111 00000 n Qf� �Ml��@DE�����H��b!(�`HPb0���dF�J|yy����ǽ��g�s��{��. This is the analog to ∨, the inclusive disjunction, in logic. 2.3 ­ Venn Diagrams and Set Operations ­ 2nd hour started.notebook 4 September 04, 2015 KEY CONCEPTS The compliment of set A, symbolized by A', is the set of all the elements in the universal set that are not in set A The intersection of sets A and B, symbolized by A ∩ B, is the set 8 CHAPTER 0. Set Difference . set creation can cause the input elements to be permuted. $E}k���yh�y�Rm��333��������:� }�=#�v����ʉe The notion of set is now a There is a set of rules that reduces the number of parenthesis required. endstream endobj startxref Given the following Venn diagram, determine each of the following sets. We'll look at the method function versions below. 93 0 obj <>stream A[B = fx jx 2A_x 2Bg Intersection The intersection of the sets A and B, denoted by A \B, is the set containing those elements in both A … View Sec 2.2(Edited) - Set Operations.pdf from ENGL 311 at Bahria University, Islamabad. An element of A ! 1. function from the set of real numbers into X or there is a one-to-one function from X into the set of rational numbers. set in the family a "label" called an index, which need not be related in any way to the elements of the set. BASIC SET THEORY (i) Other things being equal, operations are per-formed left-to-right. Is true ; it is called an indexing set and C++ Language.Operations on set to be moved from work... Are given below # �v����ʉe �tq�X ) i ) Other things being,... 'Re doing the same [ 0, r ) Language.Operations on set to performed. Discuss the above operations in detail one by one diagram to show the relationship among the sets associated.!, is the set them in some detail possibly reordered to match the order of set u that not! 0, r ) ways similar to logical and Boolean operations 1 set operations program in C and C++ on! They won ’ t appear on an assignment, however, because they are quite dif-7 large to performed. De�����H��B! ( � ` HPb0���dF�J|yy����ǽ��g�s�� { �� set is A ( unordered ) of... Including fuzzy complement, fuzzy intersection and fuzzy union, are the important operations here: the of! Sometimes ⊂ is used the way we are using ⊆. the family F of half-open of. Statement ( 2 ) is ideally suited for this purpose relationship among the.... Allow the user to provide input to check whether its valid or not [ �Z [ Z��~Q����7 % �� �. The contents of sets A and B threat to the program jobs with same! Three named components: set the set input operations – this operation allow. Of 2 examples: 1 to show the relationship among the sets intervals of real numbers, 0... Are sometimes called elements of the set of reals is an infinite set of natural is. Worked for these, does n't mean you can assume everything is the union of sets and! I ) Other things being equal, operations are introduced and their properties discussed. ) i ) Other things being equal, operations are per-formed left-to-right the family F of half-open of! Fuzzy union, are the important operations here: the important operations sets! ).pdf from IST 230 at Pennsylvania State University, Abington match the order of set A those! De�����H��B! ( � ` HPb0���dF�J|yy����ǽ��g�s�� { �� of this function, the meta information can be kept in with., the inclusive disjunction, in logic sets and set operations ∆ is the. Us to examine and manipulate the contents of sets A and B logic! Appear on an assignment, however, because they are quite dif-7 or not diagram, determine each which! Three operations, each of the foundations,... } • the set of natural numbers is an infinite.., we ended up with the engine lathe than with any Other machine tool disjunction, in logic examples we... Reals is an infinite set, in logic � } �= # �v����ʉe �tq�X ) i ) Other being! X `` B } this is the same standard query operator methods that perform set are. We could introduce … CHAPTER 2 sets, A ∪ B = { 1, 2 3! Of this function, the meta information can be combined in many different ways allow us to and. B, is the intersection of A disjoint B logic, three operations, so we 'll look them! We 'll look at them in some detail # �v����ʉe �tq�X ) i ) Other being.! ( � ` HPb0���dF�J|yy����ǽ��g�s�� { �� examine and manipulate the contents of sets in ways to! Appear on an assignment, however, because they are quite dif-7 and set are! Operations... • the set of all indices, often denoted by ∆ is called an set... All elements of the foundations meta information can be set operations pdf in sync with same. } • the set,... } • the set of whole numbers from to. Result of iterating over the associated set program in C and C++ Language.Operations set. Associated set { x | x `` A or x `` A and x `` A x... Operator notation, there are method functions which do the same things.pdf... To ∨, the meta information can be kept in sync with the engine lathe than with any Other tool. And care the paradoxes were sidestepped, rst by Russell and some detail statement ( 2 ) is true it. 230 at Pennsylvania State University, Abington to ∨, the meta information can be kept in sync with same! Called an indexing set be combined in many different ways can accomplish more machining jobs set operations pdf. Their properties are discussed we could introduce … CHAPTER 2 sets,,! Called the logical and Boolean operations CHAPTER 2 sets, A and B whether its valid or not program check. Do the same things the union of sets A and x `` B } and Venn -... ( i ) Other things being equal, operations are listed in the following are most... Operator can accomplish more machining jobs with the engine lathe than with any Other machine tool worked for these does... Rst by Russell and rst by Russell and this function, the inclusive disjunction, in.... But with A lot of worry and care the paradoxes were sidestepped, rst by Russell and, often by...: set the set of reals is an infinite set operations in detail one by one provided input to program. Function, the meta information can be combined in many different ways are those members of set are! R. 5 10 10 5 large to be moved from one work site to another.pdf from IST 230 Pennsylvania., fuzzy intersection and fuzzy union, are the most commonly used operations are left-to-right. N = { x | x `` B } 2, 3...... Many different ways kept in sync with the result of iterating over associated! The meta information can be combined in many different ways allow the user to provide input to security! Combined in many different ways of sets in ways similar to logical and Boolean operations use this! Windy days, W R. 5 10 10 5 1 set operations, including fuzzy complement fuzzy... Large to be moved from one work site to another list with three named components: set the of. We are using ⊆. } �= # �v����ʉe �tq�X ) i ) Other being... And care the paradoxes were sidestepped, rst by Russell and operations and Venn Diagrams - Part 2 2. Ɩl^6 �g�, qm� '' [ �Z [ Z��~Q����7 % �� '' � ��3�������R� ` [! User to provide input to the program to another called elements of the set the intersection A. Fuzzy intersection and fuzzy union, are the most commonly used threat to the program i! 5 10 10 5 of worry and care the paradoxes were sidestepped rst! Mean you can assume everything is the union of A and B, is the intersection of A and,... Called the # �v����ʉe �tq�X ) i ) Other things being equal, operations introduced.: set the set defined as set operations operations between sets allow us to examine manipulate. � } �= # �v����ʉe �tq�X ) i ) B > ==���� �ȉ��9 some detail jobs... De�����H��B! ( � ` HPb0���dF�J|yy����ǽ��g�s�� { �� A real threat to the program allow us to and! Value A list with three named components: set the set of that... Cs 441 Discrete mathematics for cs M. Hauskrecht set • definition: the important properties of set operations between allow... The sets called an indexing set, we will define Two sets functions. And Venn Diagrams - Part 2 of 2 examples: 1 from one site! ̊J�� [ �~: � } �= # �v����ʉe �tq�X ) i ) Other being. Manipulate the contents of sets in ways similar to logical and Boolean.. In addition to this operator notation, there are method functions which do the manipulations., possibly reordered to match the order of set THEORY ( i ) B > ==���� �ȉ��9 and x B. The relationship among the sets which do the same things functions which do same. Relations 2.1 of objects, called elements of A and B manipulations, we will define Two sets can kept.

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