discreet math cardinality presentation data

8/9/2019 discreet math CARDINALITY presentation data

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CARDINALITY:

Definition:In mathematics, the cardinality of a set is a measure of the "numberof elements of the set". For example, the set A = {2, 4, 6} contains 3elements, and therefore A has a cardinality of 3. There are two approaches tocardinality one which compares sets directly using bijections,and injections, and another which uses cardinal numbers. The cardinality ofa set A is usually denoted |A|, with a vertical bar on each side; this is thesame notation as absolute value and the meaning depends on context.

Alternately, the cardinality of a setA

may be denoted by or #A

.Comparing sets:

Case 1: |A| = |B|

Two sets A and B have the same cardinality if there exists a bijection,that is, an injective and surjective function, from A to B.

For example, the set E= {0, 2, 4, 6, ...} of non-negative evennumbers has the same cardinality as the set N = {0, 1, 2, 3, ...}

of natural numbers, since the functionf(n) = 2n is a bijectionfrom N to E.

Case 2: |A| |B|

A has cardinality greater than or equal to the cardinality ofB if thereexists an injective function from B into A.

Case 3: |A| > |B|

A has cardinality strictly greater than the cardinality ofB if there is an

injective function, but no bijective function, fromB

toA

.For example, the set Rof all real numbers has cardinality strictlygreater than the cardinality of the set N of all natural numbers,

because the inclusion map i : N R is injective, but it can be shownthat there does not exist a bijective function from N to R.


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Cardinality - Cardinal numbers

Above, "cardinality" was defined functionally. That is, the "cardinality" of aset was not defined as a specific object itself. However, such an object can

be defined as follows.The relation of having the same cardinality is called equinumerosity, andthis is an equivalence relation on the class of all sets. The equivalenceclass of a set A under this relation then consists of all those sets which havethe same cardinality as A. There are two ways to define the "cardinality of aset":

1.The cardinality of a set A is defined as its equivalence class under

equinumerosity.2.A particular class of representatives of the equivalence classes isspecified. The most common choice is the Von Neumann cardinalassignment. This is usually taken as the definition of cardinal numberin axiomatic set theory.

The cardinalities of the infinite sets are denoted

The cardinality of the natural numbers is denoted aleph-null (0), while

the cardinality of the real numbers is denoted by c, and is also referredto as the cardinality of the continuum. We can show that c = 20; thisalso being the cardinality of the set of all subsets of the natural numbers.The continuum hypothesis says that 1 = 20, i.e. 20 is the smallestcardinal number bigger than 0, i.e. there is no set whose cardinality isstrictly between that of the integers and that of the real numbers.

Cardinality - Countable and uncountable sets

Assuming the axiom of choice holds, the law of trichotomy holds forcardinality, so we have the following definitions.

y Any set with cardinality less than that of the natural numbers is said tobe a finite set.


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y Any set that has the same cardinality as the set of the natural numbersis said to be a countable infinite set.

y Any set with cardinality greater than that of the natural numbers issaid to be uncountable.

Cardinality - Examples and other properties

y If, for instance, set X is defined as X = {a, b, c}, and set Y is definedas Y = {apples, oranges, peaches}, then | X | = | Y | because they bothhave three elements.

y If for two sets X and Y, | X | | Y | , then there exists a set Z as asubset of Y such that | X | = |Z | .

Such a property allows for the comparison of how many elements are

contained in two or more sets without resorting to an intermediate set (viz.the natural numbers).

y Within the realm of uncountable sets, there exists a class of sets Ysuch that | Y | = c(cardinality of set of real numbers). Such sets aresaid to have "cardinality of the continuum."

y It can be proven that there exists no set X such that for any set Y, | Y | |X | .


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However, the 5 students who are members of both clubs are counted twice inthis addition; thus we must subtract 5 from the total, to arrive at 10 + 12 5= 17 as the number of students in the union of the art and music clubs.The preceding example demonstrates the general formula for the cardinalityof the union of two sets A and B (when A and B might intersect),

|A B| = |A| + |B| |A B|.

In adding the number of elements in A to the number in B, we count thoseelements common to A and B twice; these are the elements in theintersection AB. To correct for this double-counting we must subtract fromour sum the number of elements in AB

EXAMPLE 2

In an ordinary deck of playing

cards, how many cards are?(a) Either red cards or face cards?(b) Either spades or queens?

For the benefit of those not familiarwith playing cards, at the right isdisplayed a standard deck of 52 cards.There are foursuits of 13 cards each,called spades(), hearts(), diamond

s(), and clubs(). The spades andclubs are black, and the hearts anddiamonds red. Each suit hasten numericalcards, labeled ace, 2, 3,4, 5, 6, 7, 8, 9, 10, as well asthreeface cards, called jack, queen,and king. (The one card is called anace; in some games it is consideredthe highest card in the suit, and inother games the lowest.) Altogether

there are 26 black cards, 26 red cards,12 face cards, and 40 numerical cards.

(a) Let R denote the set of redcards, and F the set of face cards. Then

|R| = 26 , |F| = 12 .We are interested in the number ofcards in the union R F. It is tempting to add 26 and 12 and conclude that

Kings :

Queens :

Jacks :

Numer-

ical

Cards :

Spds. Hts. Diams. Clubs


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there are 38 cards that are either red or face cards. However, the sets R and Fare not disjoint; in fact the intersection R F contains 6 cards which are

both red cards and face cards, and these have been counted twice. Therefore,the number of cards that are either red or face cards is

|R F| = |R| + |F| |R F| = 26 + 12 6 = 32.(b) Let S denote the spades and Q the queens. Then |S| = 13, |Q| = 4, |S Q|

= 1, and|S Q| = |S| + |Q| |S Q| = 13 + 4 1 = 16.

(The queen of spades gets counted twice in adding the spades and thequeens.)Any set S is disjoint from its complement ~S, while the union of S and ~S isthe universal set U. Consequently, if we add the number of elements in S tothe number not in S, we get the total number of elements in U - that is,

|S| + |~S| = |U|.

We illustrate this basic formula with a simple example.

EXAMPLE 3

In Susan's refrigerator there are 9 eggs, of which 3 arerotten. How many of the eggs are good? The universeU, consisting of all the eggs, has 9 members, whilethe subset R of rotten eggs has 3 members. The goodeggs make up the complement, ~R. Therefore,

|R| + |~R| = |U| ,|~R| = |U| |R| = 9 3 = 6 .

There are 6 good eggs.One can solve counting problems involving unions and intersections of setsalso with the help of Venn diagrams. This method is perhaps even preferred,

because no formulas are needed and usually in the end we wind up withmore information than was initially sought. The next several examplesdemonstrate the Venn diagram method.

EXAMPLE 4Among the voters at a neighborhood board meeting there are 10 females,

23 Democrats, and 7 female Democrats. How many voters are either femaleor Democrat?


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First we work the problem with our formula. Let F be the set of femalesat the meeting and D the set of Democrats. We are given that |F| = 10, |D| =23, and |F D| = 7. The number of voters who are female or Democrat thenis

|F D| = |F| + |D| |F D| = 10 + 23 7 = 26

Cardinality ofreal number

In mathematics, the cardinality of the continuum, sometimes also

called the power of the continuum, is the size (cardinality) of the set

of real numbers (sometimes called the continuum). The cardinality

of is denoted by or by the symbol (a lowercase Fraktur letter

C). As a cardinal number, is equal to Beth one ( ). If

the continuum hypothesis holds, then is also equal to Aleph one (

).Georg Cantor showed that the cardinality of the continuum is larger

than that of the set of natural numbers , namely where

(aleph-nought) denotes the cardinality of . In other words,

although and are both infinite sets, the real numbers are in some

sense "more numerous" than the natural numbers.


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General Description

In mathematics, the cardinality of a set is a measure of the"number of elements of the set". For example, the set A 2, 4,6 contains 3 elements, and therefore A has a cardinality of 3.There are two approaches to cardinality one which comparessets directly using bijections and injections, and anotherwhich uses cardinal numbers.

The cardinality of a set A is usually denoted A , with avertical bar on each side this is the same notation as absolutevalue and the meaning depends on context. Alternately, thecardinality of a set A may be denoted by or A.

Above, "cardinality" was defined functionally. That is, the"cardinality" of a set was not defined as a specific objectitself. However, such an object can be defined as follows.

The relation of having the same cardinality is calledequinumerosity, and this is an equivalence relation on theclass of all sets. The equivalence class of a set A under thisrelation then consists of all those sets which have the same

cardinality as A. There are two ways to define the"cardinality of a set"

discreet math cardinality presentation data

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