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Union (set theory)

In set theory, the union (denoted as ∪) of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.

Definition

The union of two sets A and B is the collection of points which are in A or in B (or in both):

A \cup B = \{ x: x \in A \,\,\,\textrm{ or }\,\,\, x \in B\}

A simple example:

A = \{1,2,3,4\}
B = \{5,6,7,8\}
A \cup B = \{1,2,3,4,5,6,7,8\}

Other more complex operations can be done including the union, if the set is for example defined by a property rather than a finite or assumed infinite enumeration of elements. As an example, a set could be defined by a property or algebraic equation, which is referred to as a solution set when resolved. An example of a property used in a union would be the following:

A = {x is an even number, x > 1}
B = {x is an odd number, x > 1}
A \cup B = \{2,3,4,5,6,...\}

If we are then to refer to a single element by the variable "x", then we can say that x is a member of the union if it is an element present in set A or in set B, or both.

Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents. The number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.

Algebraic properties

Binary union is an associative operation; that is,

A∪ (B∪ C) = (A∪ B) ∪ C.

The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e. either of the above can be expressed equivalently as A∪ B∪ C). Similarly, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union. That is, A∪ {} = A, for any set A. In terms of the definitions, these facts follow from analogous facts about logical disjunction.

Together with intersection and complement, union makes any power set into a Boolean algebra. For example, union and intersection distribute over each other, and all three operations are combined in De Morgan's laws. Replacing union with symmetric difference gives a Boolean ring instead of a Boolean algebra

Forms

Finite unions

More generally, one can take the union of several sets at once. The union of A, B, and C, for example, contains all elements of A, all elements of B, and all elements of C, and nothing else. Formally, x is an element of A∪ B∪ C if and only if x is in A, B, or C.

Union is an associative operation, it doesn't matter in what order unions are taken. In mathematics a finite union means any union carried out on a finite number of sets: it doesn't imply that the union set is a finite set.

Arbitrary unions

The most general notion is the union of an arbitrary collection of sets. If M is a set whose elements are themselves sets, then x is an element of the union of Mif and only iffor at least one element A of M, x is an element of A. In symbols:

x \in \bigcup\mathbf{M} \iff \exists A \in \mathbf{M},\ x \in A.

That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union in axiomatic set theory.

This idea subsumes the above paragraphs, in that for example, A∪ B∪ C is the union of the collection {A,B,C}. Also, if M is the empty collection, then the union of M is the empty set. The analogy between finite unions and logical disjunction extends to one between arbitrary unions and existential quantification.

The notation for the general concept can vary considerably, such as the following:

\bigcup \mathbf{M},
\bigcup_{A\in\mathbf{M}} A.
\bigcup_{i\in I} A_{i},

which refers to the union of the collection {Ai : i is in I}. Here I is an index set, and Ai is a set for every i in I. In the case that the index set I is the set of natural numbers, the notation is analogous to that of infinite series:

\bigcup_{i=1}^{\infty} A_{i}.

When formatting is difficult, this can also be written "A1∪ A2∪ A3∪ ···". (This last example, a union of countably many sets, is very common in analysis; for an example see the article on σ-algebras.) Whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size.

Intersection distributes over infinitary union, in the sense that

A \cap \bigcup_{i\in I} B_{i} = \bigcup_{i\in I} (A \cap B_{i}).

Infinitary union can be combined with infinitary intersection to get the law

\bigcup_{i\in I} \bigg(\bigcap_{j\in J} A_{i,j}\bigg) \subseteq \bigcap_{j\in J} \bigg(\bigcup_{i\in I} A_{i,j}\bigg).

Intersection (set theory)

In mathematics, the intersection (denoted as ∩) of two setsA and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

For explanation of the symbols used in this article, refer to thetable of mathematical symbols.

Basic definition

The intersection of A and B is written "A∩ B". Formally:

x∈ A∩ Bif and only if
* x∈ Aand
* x∈ B.
For example:
* The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
* The number 9 is not in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of odd numbers {1, 3, 5, 7, 9, 11, …}.

If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A∩ B = âˆ…. For example the sets {1, 2} and {3, 4} are disjoint, written
{1, 2} ∩ {3, 4} = âˆ….

More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A∩ B∩ C∩ D = A∩ (B∩ (C∩ D)). Intersection is an associative operation; thus,
A∩ (B∩ C) = (A∩ B) ∩ C.

If the sets A and B are closed under complement then the intersection of A and B may be written as the complement of the union of their complements, derived easily from De Morgan's laws:
A∩ B = (Ac∪ Bc)c

Arbitrary intersections

The most general notion is the intersection of an arbitrary nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, x is an element of A. In symbols:

\left( x \in \bigcap \mathbf{M} \right) \leftrightarrow \left( \forall A \in \mathbf{M}, \ x \in A \right).

The notation for this last concept can vary considerably. Set theorists will sometimes write "M", while others will instead write "A∈M A". The latter notation can be generalized to "i∈I Ai", which refers to the intersection of the collection {Ai : i âˆˆ I}. Here I is a nonempty set, and Ai is a set for every i in I.

In the case that the index setI is the set of natural numbers, notation analogous to that of an infinite series may be seen:

\bigcap_{i=1}^{\infty} A_i.

When formatting is difficult, this can also be written "A1 âˆ© A2 âˆ© A3 âˆ© ...", even though strictly speaking, A1 âˆ© (A2 âˆ© (A3 âˆ© ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for an example see the article on σ-algebras.)

Finally, let us note that whenever the symbol "∩" is placed before other symbols instead of between them, it should be of a larger size ().

Nullary intersection

Note that in the previous section we excluded the case where M was the empty set (). The reason is as follows. The intersection of the collection M is defined as the set (see set-builder notation)

\bigcap \mathbf{M} = \{x : \forall A \in \mathbf{M}, x \in A\}.

If M is empty there are no sets A in M, so the question becomes "which x's satisfy the stated condition?" The answer seems to be every possible x. When M is empty the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set, which according to standard (ZFC) set theory, does not exist.

A partial fix for this problem can be found if we agree to restrict our attention to subsets of a fixed set U called the universe. In this case the intersection of a family of subsets of U can be defined as

\bigcap \mathbf{M} = \{x \in U : \forall A \in \mathbf{M}, x \in A\}.

Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-defined set by assumption.


Power set

In mathematics, the power set (or powerset) of any setS, written \mathcal{P}(S), P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.

Any subsetF of \mathcal{P}(S) is called a family of setsover S.

Example

If S is the set {x, y, z}, then the subsets of S are:

  • {} (also denoted \varnothing, the empty set)
  • {x}
  • {y}
  • {z}
  • {x, y}
  • {x, z}
  • {y, z}
  • {x, y, z}

and hence the power set of S = \left\{x, y, z\right\} is

\mathcal{P}(S) = \left\{\{\}, \{x\}, \{y\}, \{z\}, \{x, y\}, \{x, z\}, \{y, z\}, \{x, y, z\}\right\}\,\!.

Properties

If S is a finite set with |S| = n elements, then the power set of S contains |\mathcal{P}(S)| = 2^n elements.

Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. For example, the power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see cardinality of the continuum).

The power set of a set S, together with the operations of union, intersection and complement can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras this is no longer true, but every infinite Boolean algebra is a subalgebra of a power set Boolean algebra (see Stone's representation theorem).

The power set of a set S forms an Abelian group when considered with the operation of symmetric difference (with the empty set as its unit and each set being its own inverse) and a commutativemonoid when considered with the operation of intersection. It can hence be shown (by proving the distributive laws) that the power set considered together with both of these operations forms a commutative ring.

In set theory, XY is the set of all functions from Y to X. As 2 can be defined as {0,1} (see natural number), 2S (i.e., {0,1}S) is the set of all functions from S to {0,1}. By identifying a function in 2S with the corresponding preimage of 1, we see that there is a bijection between 2S and \mathcal{P}(S), where each function is the characteristic function of the subset in \mathcal{P}(S) with which it is identified. Hence 2S and \mathcal{P}(S) could be considered identical set-theoretically. (Thus there are two distinct notational motivations for denoting the power set by 2S: the fact that this function-representation of subsets makes it a special case of the XY notation and the property, mentioned above, that |2S| = 2|S|.)

This notion can be applied to the example above to see the isomorphism with the binary numbers from 0 to 2n-1 with n being the number of elements in the set. In S, a 1 in the position corresponding to the location in the set indicates the presence of the element. So {x, y} = 110.

For the whole power set of S we get:

  • { } = 000 (Binary) = 0 (Decimal)
  • {x} = 100 = 4
  • {y} = 010 = 2
  • {z} = 001 = 1
  • {x, y} = 110 = 6
  • {x, z} = 101 = 5
  • {y, z} = 011 = 3
  • {x, y, z} = 111 = 7

Relation to binomial theorem

The power set is closely related to the binomial theorem. The number of sets with k elements in the power set of a set with n elements will be a combination C(n,k), also called a marketing mix. An organization or set of organizations (go-betweens) involved in the process of making a product or service available for use or consumption by a consumer or business user.

The other three parts of the marketing mix are product, pricing, and promotion.

The distribution channel

Distribution is also a very important component of Logistics & Supply chain management. Distribution in supply chain management means the distribution of good from one business to other it can be factory to supplier, supplier to retailer or retailer to the end customer. It is defined as a chain of intermediaries, each passing the product down the chain to the next organization, before it finally reaches the consumer or end-user. This process is known as the 'distribution chain' or the 'channel.' Each of the elements in these chains will have their own specific needs, which the producer must take into account, along with those of the all-important end-user.

Channels

A number of alternate 'channels' of distribution may be available:

  • Distributor, who sells to retailers,
  • Retailer (also called dealer or reseller), who sells to end customers
  • Advertisement typically used for consumption goods

Distribution channels may not be restricted to physical products alice from producer to consumer in certain sectors, since both direct and indirect channels may be used. Hotels, for example, may sell their services (typically rooms) directly or through travel agents, tour operators, airlines, tourist boards, centralized reservation systems, etc. process of transfer the products or services from Producer to Customer or end user.

There have also been some innovations in the distribution of services. For example, there has been an increase in franchising and in rental services - the latter offering anything from televisions through tools. There has also been some evidence of service integration, with services linking together, particularly in the travel and tourism sectors. For example, links now exist between airlines, hotels and car rental services. In addition, there has been a significant increase in retail outlets for the service sector. Outlets such as estate agencies and building society offices are crowding out traditional grocers from major shopping areas.

Channel decisions

Channel Sales is nothing but a chain for to market a product through different sources.

  • Channel strategy
  • Gravity & Gravity
  • Push and Pull strategy
  • Product (or service)
  • Cost
  • Consumer location

Managerial concerns

The channel decision is very important. In theory at least, there is a form of trade-off: the cost of using intermediaries to achieve wider distribution is supposedly lower. Indeed, most consumer goods manufacturers could never justify the cost of selling direct to their consumers, except by mail order. Many suppliers seem to assume that once their product has been sold into the channel, into the beginning of the distribution chain, their job is finished. Yet that distribution chain is merely assuming a part of the supplier's responsibility; and, if they have any aspirations to be market-oriented, their job should really be extended to managing all the processes involved in that chain, until the product or service arrives with the end-user. This may involve a number of decisions on the part of the supplier:

  • Channel membership
  • Channel motivation
  • Monitoring and managing channels
  1. Intensive distribution - Where the majority of resellers stock the 'product' (with convenience products, for example, and particularly the brand leaders in consumer goods markets) price competition may be evident.
  2. Selective distribution - This is the normal pattern (in both consumer and industrial markets) where 'suitable' resellers stock the product.
  3. Exclusive distribution - Only specially selected resellers or authorized dealers (typically only one per geographical area) are allowed to sell the 'product'.

Channel motivation

It is difficult enough to motivate direct employees to provide the necessary sales and service support. Motivating the owners and employees of the independent organizations in a distribution chain requires even greater effort. There are many devices for achieving such motivation. Perhaps the most usual is `incentive': the supplier offers a better margin, to tempt the owners in the channel to push the product rather than its competitors; or a compensation is offered to the distributors' sales personnel, so that they are tempted to push the product. Julian Dent defines this incentive as a Channel Value Proposition or business case, with which the supplier sells the channel member on the commercial merits of doing business together. He describes this as selling business models not products.

Monitoring and managing channels

In much the same way that the organization's own sales and distribution activities need to be monitored and managed, so will those of the distribution chain.

In practice, many organizations use a mix of different channels; in particular, they may complement a direct salesforce, calling on the larger accounts, with agents, covering the smaller customers and prospects. These channels show marketing strategies of an organisation. Effective management of distribution channel requires making and implementing decision in these areas.



From Yahoo Answers

Question:I have a couple problems I cannot really figure out Let S be a set with an associative law of composition and with an identity elemment. Prove that the subset of S consisting of invertible elements is a group. Let S be any set. Prove that the law of composition defined by a*b = a is associative. and last one Suppose that G is a group in which all elements x satisfy x^2 = e. Prove that G is a commutative group. (e is the identity element) I know they are probably simple, but I do not know much about the associative law of composition, because for whatever reason its not in the book. Thanks. i know associativity, but in the context its given i dont understand what it is asking to show.

Answers:"Associative law of compostion" simply means that the operation is composition, and that the set is associative; and associative means (a*b)*c = a*(b*c), where * represents the operation. If you have any more questions, ask me.

Question:Can anyone show me how to prove the absorption laws, that is A U (A ^ B) = A A ^ (A U B) = A where ^ means intersection, using only the fundamental laws of set algebra? thanks! @asja2097: Why can we assume AUB = u?? (Third line)

Answers:A U (A^B) = (AUA) ^ (AUB) (associative law) = A ^ (AUB) (see below) = A ^ U (may assume AUB = U) = A (by identity law for intersection)

Question:i need help with these asap. I usually contribute answers so if people can help me out here i would greatly appreciate it Set theory 1) If g: a->b and f: g(A) -> C are injective functions, prove that f composition g is also one to one 2) Define a countable set and show that the set of natural #s is countable 3) Define an equivilance relation, and for a,b,c,d exists in Z define a/b equiv c/d if and only if ad=bc. Show that tilde is an equivalence relationship 4) what is the cardinality of the set {1/1, 1/2, 1/3 ....} .. prove it using 1 to 1 correspondence awesome answer thanks

Answers:1. Something like, suppose that f(g(x))=t and f(g(y))=t. Then, as f is 1-1, g(x)=g(y). Then, as g is 1-1, x=y. 2. A Countable Set A is a set in which there exists a function f, such that f is injective, and f maps A to the naturals. The naturals are countable by the identity map. 3) An equivilence relation is a relation which is reflexive (for all x, x~x), symmetric ( for all x,y (x~y -> y~x ) ) and transitive (for all x,y,z ( (x~y and y~z) -> x~z ) ). Clearly this relation is reflextive, because ad = da. Clearly this symmetric, because if ad=bc, then cb=da. Clearly this is transitive, because if a/b~c/d then ad=bc. And if c/d~e/f, cf=ed. Then af=be (just by the way rationals will work) 4) This set is countable infinity. let f(a) = 1/a map the naturals to the set. This is bijective, ie. It is clearly 1-1 and onto, thus this set has the same size as N.

Question:these 3 questions in my summer "fun" packet are not addressed in the reading material they gave us, so cud u plz help me out =) Explain the difference between a law and a theory. Explain the difference between qualitative and quantitative data. Is an experiment designed to determine the identity of a compound by its color quantitative or qualitative? (my spidy sense tells me the answer to second part is qualitative) Define precision and accuracy. Which one implies the other for a set of data?

Answers:Ghandi is incorrect with regards to laws and theories. It's easier to look at these things from the bottom of the scientific method to the top. First a scientist has a hypothesis - "If I throw rocks at Stan, then stan will be angry." This is called an "if/then statement," and that's what a hypothesis is. Theory: An expectation based on many repeated tests of a hypothesis. "I have thrown rocks at Stan many times, and Stan has gotten angry each time. Therefore, Stan will get angry when I throw rocks." Theories have been shown to be true by repeated testing. Law: An expectation that is almost ALWAYS true, regardless of how you look at a situation - it is more general. Theories can become laws if they become general enough. "If I throw rocks at a person, that person will get angry. This is the Law of Rock Throwing." As for data: "Qualitative" means you've seen it but haven't measured it. "I threw some rocks at Stan" is a qualitative statement. "Quantitative" means that something is countable. "I threw 27 rocks at Stan." Blue is qualitative. The SHADE of blue is quantitative. Precision: Always coming close to the same answer every time. If I throw 10 rocks at Stan, and they always hit his left hand, then I am a precise thrower. Accuracy: Where the answer actually is. If I was throwing rocks at Stan's right foot, but kept hitting his left hand, I was not an accurate thrower. You want data to be both accurate and precise.

From Youtube

Set Theory :Learning Outcomes: As watching the illustration students or learners will be able to Define sets and subsets properties by using the live notation

Dan Says - Set Theory: The Basics :Hello everybody :), Upon request I have made this video to explain the basic concepts of Set Theory: -definition of set -difference from multiset -union -intersection -set subtraction -empty set -subset, superset -equals -DeMorgans Law Enjoy :) Have a beautiful day!