Let A be a set in a universal set U, and let A 1 be a complement for A. If A 2 is another complement for A, we have: A 1 = U ∩ A 1 = ( A ∪ A 2) ∩ A 1 = ( A ∩ A 1) ∪ ( A 2 ∩ A 1) = A 2 ∩ A 1. Similarly, A 2 = U ∩ A 2 = ( A ∪ A 1) ∩ A 2 = ( A ∩ A 2) ∪ ( A 1 ∩ A 2) = A 1 ∩ A 2. Hence, A 1 = A 2 ∩ A 1 = A 1 ∩ A 2 = A 2 If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U , either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U : [4 Find the complement of A in U A = { x / x is a number bigger than 4 and smaller than 8} U = { x / x is a positive number smaller than 7} A = { 5, 6, 7} and U = { 1, 2, 3, 4, 5, 6} A c = { 1, 2, 3, 4} Or A c = { x / x is a number bigger than 1 and smaller than 5 } The graph below shows the shaded region for the complement of set A what I want to do in this video is introduce the idea of a universal set or the universe that we care about and also the idea of a complement or an absolute complement and so if we're doing it as a Venn diagram the universe is usually depicted as some type of a rectangle right over here and it itself is a set and it usually is denoted with the capital u you for universe not to be confused with the union set notation and you could say that the universe is all possible things that could be in.
In set theory and other branches of mathematics can have the two kinds of complements are defined, the relative complement and the absolute complement. Given a set A, then also be a complement of A is the set of all elements in the universal set U,but not in A. We can write A c You can also say complement of A in U . Types of Complements . Relative complement So this part is Oh, so it means that a union in percent we'll have elements, I said. Old elements will be in L A. So this means that a union, uh, the cassette movie sets itself so embarked me. We have a section u Universal side. So imagine that we have this universal that complains or the els. And we have said so. This is the universe aside. It contains every element in any open set. Since A is a subset off the university sent, we can say that intersection you will be sent a One generally says that sets are represented by circles; but these sets can only be subsets of U. The complement of a set A is then given by that portion of the rectangle outside of A's circle. Strictly speaking, this is the relative complement U \ A of A relative to U; but in a context where U is the universe, it can be regarded as the absolute complement A C of A Let A,B,C be subsets of the universal set U. If n(U) =692,n(B) = 230,n(C) =370,n(B ∩C) = 20, n(A ∩B′ ∩ C ′) =10 ,find n(A′ ∩B ∩ C ′)
Notice that the definition of set union tells us how to form the union of two sets. It is the associative law that allows us to discuss the union of three sets. Using the associate law, if A, B, and C are subsets of some universal set, then we can define A ∪ B ∪ C to be (A ∪ B) ∪ C or A ∪ (B ∪ C) (3 points) Let A and B be sets with a universe U. Prove that if Ac Bc, then P(B) P(A) 3. ANSWER QUIZ 1 : MAT1202 SET THEORY (SEC2) TOPIC Axiom of Equality, Operation & Power set Subgroups SCORE 10 points QUIZ TIME Tue 25 Jan 2017, 3th Week, Semester 2/2016 TEACHER Thanatyod Jampawai, Ph.D., Faculty of Education, Suan Sunandha Rajabhat University 1. (3 points) Write out a set in builder form. Let U be a universe. Prove that for all A ⊆ U, there exists a unique set B ⊆ U, such that (A ∪ B)-(A ∩ B) = ∅. 6. The ten integers from 1 to 10 are evenly spaced in random order around a circle. Prove that there must be three integers in adjacent positions around the circle that have a sum which is greater than or equal to 17. 7. Sudokus and Kenkens are popular types of mathematical.
Often the universal set may not be explicitly stated and it may be unclear as to just what it is. At other times it will be clear. Complement of a set. The complement of a set A with respect to a given universal set U is the set of elements in U that are not in A. The complement of A is typically denoted by A c or A'. Finite and infinite sets Set Complementation If U is a universal set and A is a subset of U,then a. Uc = ? b. ?c = U c. (Ac)c = A d. A[Ac = U e. A\Ac = ? Properties of Set Operations Let U be a universal set. IfA, B,andC are arbitrary subsets of U,then A[B = B [A Commutative law for union A\B = B \A Commutative law for intersection A[(B [C)=(A[B)[C Associative law for unio Let A be a set, U its universe, Ο the empty set. The notation Ac denotes the complement Problem (a) Prove: AUA U. (b) Prove: An Ac=0. Hint: Remember, to show that two sets are equal, you should show b inelusions. Be careful showing ine in the empty set
Moreover, we improve the notion of complement of a soft set, and prove that certain De Morgan's laws hold in soft set theory with respect to these new definitions. Previous article in issue; Next article in issue; Keywords. Soft sets. Union . Intersection. Complement. Difference. 1. PreliminariesIn this section, we recall some basic notions in soft set theory. Let U be an initial universe. Example 1.5.1 If the universe is $\Z$, then $\{x:x>0\}$ is the set of positive integers and $\{x:\exists n\,(x=2n)\}$ is the set of even integers. $\square$ If there are a finite number of elements in a set, or if the elements can be arranged in a sequence, we often indicate the set simply by listing its elements Let Abe a set contained in our universe U. The complement of A, denoted AC or A, is the set: A:= fx2U: x62Ag Example 6. Let U= [5], and let A= f1;2;4g. Then A= f3;5g. De nition 6 (Set Di erence). Let A;Bbe sets contained in our universe U. The di erence of Aand B, denoted AnBor A B, is the set: AnB= fx: x2Aand x62Bg Example 7. Let U= [5], A= f1;2;3gand B= f1;2g. Then AnB= f3g. Remark: The Set. Given a particular universe U, the complement of a set x is U - x; this acts like negation. Note that the complement is not defined except in the context of some universe. 1.5. Proving things about sets. Before throwing in any more axioms, let's take a brief break to talk about strategies for proving statements about sets: 1.5.1. To prove that.
Let Ube the universe of all keys. For example, Ucould be the set of all 64 bit strings. In this case jUj= 264. This is a very large universe, but we do not need to store all of these 264 keys, we only need to store a subset SˆU. Suppose that we know that the size of the subset we will need to store is less than or equal to n The complement S c of a set S is defined to be the set of all elements of the universe U that are not elements of S. Note complement is a unary operator; that is, it is a function on a single input. In contrast, set union is a binary operator; that is, set union is a function on two inputs. Examples of languages L (sets of strings) and their complements L c. L = * L c = {}. L = {the set of. • the truth set of A(x) is not the entire universe U * [ the truth set of A(x) ] 6= U < Let B and D be any sets. Then B = D if and only if Bc = Dc.> * [ the truth set of A(x) ]c 6= Uc * [ the truth set of ∼ A(x) ] 6= ∅ • the truth set of [∼ A(x)] is nonempty <by def.: the truth set of P(x) is nonempty ⇔ the truth set of P(x.
Proof: Let abe an arbitrary even integer. Then, by definition, a= 2 b for some integer b (depending on a). Squaring both sides, we get a2= 4 b2 = 2(2 b2). Since 2b2 is an integer, by definition, a2is even. Since awas arbitrary, it follows that the square of every even number is even. Even and Odd Prove The square of every odd integer is odd. Even(x) ≡ ∃ =2 Odd(x) ≡ ∃ (=2+1. Example 1.5.1 If the universe is $\Z$, then $\{x:x>0\}$ is the set of positive integers and $\{x:\exists n\,(x=2n)\}$ is the set of even integers. $\square$ If there are a finite number of elements in a set, or if the elements can be arranged in a sequence, we often indicate the set simply by listing its elements Show that, i A and B are sets, then (A ∩ B) ∪ (A ∩ BC) = A. [Note: BC is another way of writing the complement of set B] There are two ways of solving set proofs like these, one is to lo ok at an arbitrary point and use the properties of sets t Often, the context provides a 'universe' of all possible elements pertinent to a given discussion. Suppose, we have given such a set of 'all' elements and let us call it \(U\). Then, the complement of a set \(A\), denoted by \(A^c\) , is deﬁned as \(A^c = U \setminus A.\) In the following theorem the existence of a universe \(U\) is.
Assume all sets are subsets of a universal set U. For all sets A, B, and C, A ∩ (A ∪ B) = A. Proof: Will work on this in the class, but you can find an answer at the back of the textbook. Proving Set Identities Algebraically Alternatively, we can prove set properties algebraically using the set identity laws. o Example: [Example 6.3.2. Each element that can be inserted in the set has a key which is drawn from the universe U. The subset S that is stored in our set is comparatively very small, that is jS j<<jU j. What we would like is a data structure that supports the above operations if possible in time O(1) while using only O(jS j) space. Assume now that the subset S of U.
Consider a universe U= fe 1;:::;e ngof nelements, a collection of subsets S= fS 1;:::;S mgof msubsets of Usuch that U= S m i=1 S i, and a non-negative1 cost function c: S!R+. If S i = fe 1;e 2;e 5g, then we say S i covers elements e 1, e 2, and e 5. For any subset T S, de ne the cost of Tas the cost of all subsets in T. That is, c(T) = X S i2T c(S i) De nition 1.2 (Minimum set cover problem. SETS, NUMBERS, AND PROOFS like numbers. For example, if Ak = fn 2 N: n > kg is the set of natural numbers greater than k, then you could have a set of sets, B = fA1, A10, A6g.Sets are unordered, so the previous deﬁnition of B is the same as B = fA1, A6, A10g.Also, sets do not contain duplicates, so for example, f1,1,2g f1,2g.Sets can be empty There are sets with infinite cardinality, such as \(\N\text{,}\) the set of rational numbers (written \(\mathbb Q\)), the set of even natural numbers, and the set of real numbers (\(\mathbb R\)). It is possible to distinguish between different infinite cardinalities, but that is beyond the scope of this text. For us, a set will either be infinite, or finite; if it is finite, the we can.
In Class 7, we defined set difference as: $$\forall x. x \in A - B \iff x \in A \wedge x \notin B.$$ Provide an alternate (but equivalent in meaning) definition of set difference using only the other defined set operations (you may use any of the union ($\cup$), intersection ($\cap$), and complement ($\overline{S}$) operations in your definition, but no other operations or qualifiers). A good. Proof. Let Σ = (Ω,Π), let Σ1 For the only if part assume that F is satisﬁable and let Abe a Σ-model of F. Then we deﬁne a Σ1-algebra Bin such a way that Band Ahave the same universe, fB = fA B = pAB is the identity relation on the universe. It is easy to check that Bis a model of both F˜ and of Eq(Σ). The proof of the if. Let U be a universal set, and let E be a set of parameters. Let I U denote the power set of all fuzzy subsets of U. Let A ⊆ E. A pair F, E is called a fuzzy soft set over U where F is a mapping given by F : A −→ I U . 2.8 Definition 2.7 see 12 . Let U {x1 , x2 , . . . , xn } be the universal set of elements and E {e1 , e2 , . . . , em. Sets Set Definition: A set is any collection of objects. If a set is finite and not too large, we can describe it by listing all of its elements Let's now prove that formula rigorously. In order to do so, we break up A [B into several disjoint parts. Once we've done that, we can apply our simple rule that the cardinality of a disjoint union is the sum of the cardinalities. As can be seen in gure 1, A[B can be expressed Figure 1: Union of Sets as union of three disjoint sets: the set of elements present only in A. This set can be.
Classical rough set theory has come from the need to represent subsets of a universe in terms of equivalence classes of a partition of that universe. The partition characterizes a topological space, called approximation space . K = (X, R), where . X. is a set called the universe and . R. is an equivalence relation [8, 14]. The equivalence. A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements. Say if A and B are two sets, such as A = {1,2,3} and B = {1,a,b,c}, then the universal set associated with these two sets is given by U = {1,2,3,a,b,c}. In Mathematics, the collection of elements or group of objects is. Proof: Let be an arbitrary positive integer. Another proof by Contradiction ∀ ∃ ( , ∧∀ ¬ , ∧ , ) Prove: For all positive integers there is a positive integer that is a strict multiple of and for all positive integer it is not true that is a multiple of and is a multiple of . Proof: Let be an arbitrary positive integer. Choose =2 which is a str Let U be an initial universe set and E be the set of parameters. Let IF U denote the collection of all intuitionistic fuzzy subsets of U. Let. A E pair (F, A) is called an intuitionistic fuzzy soft set over U where F is a mapping given by F: A→ IF U. 2.3 Defintion . Let F: A→ IF. U . then F is a function defined as F ( ) ={ x, ( ) ( ), ( ) ( : + where , denote the degree of membership and. On the algebraic structures of soft sets in logic 1875 Γw(2) := {T ∈ Γ(1) : T ≡ I or W (mod2)} and Γp(2) := T ∈ Γ(1) : T ≡ I, P or P2 (mod2) These equalities are deﬁned by Rankin [9]. More information can be taken here. Throughout this subsection U refers to an initial universe, E is a set of parameters, P(U) is the power set of U and A ⊂ E.Molodtsov [8] deﬁned
Let U be a finite universe, and R a n-place relation on U. Then there exist a natural number X = X(R), and equivalence relation E on U such that uniformly we have: 1. Qe and Q\~] are interpretable by Qr. 2.If\U\>ln then QR is expressible by{QE, g]1}. 3. If \U\ < Xn then every binary relation on a subset A ? U with cardinality < | U\l/2n is interpretable by QR. In case (2) of the theorem if we. A set theory textbook can cover a vast amount of material depending on the mathematical background of the readers it was designed for. Selecting the material for presentation in this book often came down to deciding how much detail should be provided when explaining concepts and what constitutes a reasonable logical gap which can be independently ﬁlled in by the reader. The initial chapters. Set Operations and Inclusion. As computer scientists, we aim to use mathematics to model computational phenomena in order to. Precisely understand how the phenomena works (usually for the purposes of implementing that phenomena as a computer program) and. Abstract differences between seemingly unrelated phenomena to discover how they are related has a model with a countable universe. Proof Let F be a formula, and let G be an equisatis able formula in Skolem form (as produced by the Normal Form transformations). Then for every set S: F has a model with universe S i G has a model with universe S. F satis able ) G satis able) G has a Herbrand model (T;I 1)) F has a model (T;I 2)) F has a countable model (Herbrand universes are countable.
Let u be the universal set and let a be a set such that a u. Not a intersect b or the complement of a intersect b means. To visualize the interaction of sets john venn in 1880 thought to use overlapping circles building on a similar idea used by leonhard euler in the 18th century. The union of two sets contains all the elements contained in either set or both sets. Shading regions with three. referred to as the universe). The complement Ac of a set A with respect to U is deﬁned by Ac = U \A = {a ∈ U |a /∈ A} We can now formulate De Morgan's laws: Proposition 1 (De Morgan's laws) Assume that A 1,A 2,...,A n are subsets of a universe U. Then (A 1 ∪A 2 ∪...∪A n)c = Ac 1 ∩A c ∩...∩Ac n and (A 1 ∩A 2 ∩...∩A n)c = Ac ∪Ac 2 ∪...∪Ac n (These rules are eas - Assume the statement holds for any set A such that |A| = n; - Let A = B ∪ {z} be an arbitrary set of n + 1 elements: ∗ Every subset of A either contains x, or does not contain x; ∗ Let P1(A) be the collection of subsets containing x; ∗ |P1(A)| = |P(B)| and |P(A) − P1(A)| = |P(B)|; ∗ |P(A)| = 2n + 2n = 2n+1. 4. Lecture 14: Set Theory Algebraic Proofs: • Universe U and its p
Let Γ be a set of linear constraints applied to the weights of portfolio P. One usual set of constraints is the long-only constraint (i.e., all weights must be positive). The portfolio, which under the set of constraints Γ max-imizes the diversiﬁcation ratio in universe U, is the Most-Diversiﬁed Portfolio, denoted as M(Γ, U) The approach taken here is to construct the natural numbers within a universe U about which we assume as little as possible. In Set Theory we said simply that U is a set whose members consist of all the objects that could possibly interest us. For our purposes here, it will be su-cient to assume simply that U is a non-vacuous family of sets that is closed under the operations of set. Set-builder notation defines a set by describing, rather than listing, its elements. Let the universe be the set of real numbers. Within this universe define a set V as follows: V = {x|x > 1.3 and x < 2π} This is read V is the set of all x such that x is greater than 1.3 and x is less than 2π. We should recognize that V is the infinite set containing all real numbers between 1.3 an Let U be a universe and S be a set such that . Let be a data structure representing S. A membership query is of the form . The response to the query is a string where , corresponding to , . An authenticated membership query is of the form . The response to the query is a string where and p is a proof for a authenticated by a CA Let be a finite collection of hash functions that map a given universe U of keys into the range {0,1, . . . , m - 1}. Such a collection is said to be universal if for each pair of distinct keys x,y U, the number of hash functions for which h(x) = h(y) is precisely
Fig : Standard complement set function C To determine the membership function of the rule, let T and H be universe of discourse of temperature and humidity, respectively, and let us define variables t ∈T and h ∈H. We represent the fuzzy terms : high, and fairly high. by A and B respectively: A = high, A ⊆T B = fairly high, B ⊆H Then the above rule can be rewritten as R(t, h): If t. Note: In these exercises the complement of A is denoted by Ac, and N = f1;2;3;:::g. 1. Which of the following sets are well de ned. [Hint: You can show a set is not well de ned if you can exhibit an object for which membership can not be determined from the speci cation.] a) The set numbers. b) The set of words in the bible. c) The set of rocks on the moon. d) The set of characters in Tolkein. Set Theoretic Operations and Fuzzy Set Properties Assignment 3 The due date for submitting this assignment has passed. As per our records you have not submitted this assignment. 1) Let and B are two fuzzy sets being given as below With the universe of discourse X — Due on 2020-02-19, 23:59 IST. { l, 2, 3, 4} . What will be the crossover 1 poin
Let U be a universe, E a set of parameters, and X a set of experts (agents). Let O be a set of opinions, Z = E × X × O and A ⊆ Z. Definition 2.1. (see ) A pair (F, A) is called a soft expert set (over U) where F is a mapping given by F : A → P (U) where P (U) denotes the power set of U. Definition 2.2 Given a universe of discourse U, a multiset can be thought of as a func- tion A4 from U to the natural numbers N. In this paper, we define a hybrid set to be any function from the universe U integers Z. These sets are called hybrid since they contain elements with either a positive or negative multiplicity For any subset A of U, the complement of A (symbolized by A′ or U − A) is defined as the set of all elements in the universe U that are not in A. Referring to the question posed above, the complement of B is everything within the universal set that is not B, including A. Before we move on, there's one more conceptual set that's quite crucial to a basic understanding: the null or empty.
many sets B is it true that (A\C) [B = C? 9. Let the universe consist of all positive integers, and let A p = fn jn is divisible by pg: Find a nonempty set C such that (f1g[A 2 [A 3 [A 5 [A 7 [A 11 [A 13 [A 17 [A 19) \C = ;: 10. Let A and B be sets with universe U. Prove (A[B)c = Ac \Bc 11. Let A and B be sets with universe U. Prove (A \B)c. 4. Let s be a positive integer. Prove that the closed interval [s;2s] contains a power of 2. fHints: Use proof by cases. Consider two cases: (a) s is a perfect power of 2, and (b) s is not a perfect power of 2, i.e. 2 k< s < 2 +1 for some integer k.g 5. Suppose A;B and C are subsets of the universe U. Prove that (a) if A B then A\C B \C
Universal hashing No matter how we choose our hash function, it is always possible to devise a set of keys that will hash to the same slot, making the hash scheme perform poorly. To circumvent this, we randomize the choice of a hash function from a carefully designed set of functions. Let U Given a universe U = fu1;:::; ung and a family of its subsets, S = fS1;:::;Skg µ P(U), S Sj2S Sj = U, set cover is the problem of ﬂnding a minimal sub-family S of S that covers the whole universe, S Sj2S Sj = U. Set cover is a classic NP-hard combinatorial optimization problem, and it is known that it can be approximated to within lnn¡lnlnn+£(1) [9,5,10]. By [6,2] it follows that.
1 r-Union-free families of sets We generalize the deﬁnition of an r-union-free family Fgiven in the book [Juk11, Section 8.6] to the case where no set in Fintersects much with the union of r other sets from F: Deﬁnition 1. Let Fbe a family of sets over the universe [n], r 1 an integer, and 2(0;1]. The family i It is important to remember that these operations (union, intersection, complement, and difference) on sets produce other sets. Don't confuse these with the symbols from the previous section (element of and subset of). \(A \cap B\) is a set, while \(A \subseteq B\) is true or false. This is the same difference as between \(3 + 2\) (which is a number) and \(3 \le 2\) (which is false) A Herbrand model H of Ghas universe the set of ground terms U H= fa;b;cg. The constants are interpreted \as themselves, i.e., we have a H= a, b H= band c H= c. Thus to specify H it remains to say how to interpret the predicate symbol P. We can represent the possibilities in the following truth table, each line of which represents a Herbrand structure. 2. P(a) P(b) P(c) G 0 0 0 0 0 0 1 1 0 1 0. Let U be an initial universe set and E be the set of parameters. Let IFU denote the collection of all intuitionistic fuzzy subsets of U. Let . A ⊆ E pair (F, A) is called an intuitionistic fuzzy soft set over U where F is a mapping given by F: A→ IFU. Definition 2.3. Let F: A→ IFU then F is a function defined as F () ={ x, ()(), ()() : } where , denote. collection of fuzzy sets over the same universe and prove DeMorgan Laws for an arbitrary collection of fuzzy sets over the same universe. 2. PRELIMINARIES H. K. Baruah [3, 4] gave an extended definition of fuzzy set in the following manner. According to him, to define a fuzzy set, two functions namely fuzzy membership function and fuzzy reference function are necessary. Fuzzy membership value.
DEFINITION :2.1[17] A pair (F,E) is called a soft set (over U) if and only if F is a mapping of E into the set of all subsets of the set U. DEFINITION :2.2[15] A soft set (F,A) over U is said to be a NULL soft set denoted by , if e A, F(e) = 3COLOR ≤ P EXACT-COVER We now reduce 3-colorability to the exact cover problem. A graph is 3-colorable iff Every node is assigned one of three colors, and No two nodes connected by an edge are assigned the same color. We will construct our universe U and sets S such that an exact covering Assigns every node in G one of three colors, and Never assigns two adjacent nodes the sam Let U be an initial universe set, E a set of parameters and A ⊆ E. Then 〈 F, A 〉 is called an intuitionistic fuzzy soft set over U where F is a mapping given by F: A → IF (U). In general, for every ε ∈ A, F (ε) is an intuitionistic fuzzy set of U and it is called intuitionistic fuzzy value set of parameter ε Proof by Universal Generalization. Universal Generalization is used for many proofs of mathematical formulas, but most less formal proofs don't state that it is used. In a proof using Universal Generalization, we pick an arbitrary value from the universe and show that the statement is true. The catch is the value must be arbitrary, showing a. Set theory - Set theory - Operations on sets: The symbol ∪ is employed to denote the union of two sets. Thus, the set A ∪ B—read A union B or the union of A and B—is defined as the set that consists of all elements belonging to either set A or set B (or both). For example, suppose that Committee A, consisting of the 5 members Jones, Blanshard, Nelson, Smith, and Hixon.
Let c be a 0-weight cycle, and let u and v be any two vertices on c. Suppose that μ * = 0 and that the weight of the path from u to v along the cycle is x. Prove that δ (s, v) = δ (s, u) + x. (Hint: The weight of the path from v to u along the cycle is -x. Let A is a given set. The membership function can be use to define a set A is given by: Operations on classical sets: For two sets A and B and Universe X: Union: This operation is also called logical OR. Intersection: This operation is also called logical AND. Complement: Difference: Properties of classical sets: For two sets A and B and Universe X: Commutativity: Associativity: Distributivity.