Cardinality of sets examples

    • [PDF File]Sets and Matrices - Northeastern University

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      Let A and B be sets. The two sets are disjoint if their intersection is the empty set. A B Cardinality of sets Inclusion-Exclusion Principle We frequently want too know how many items are in groups. Note that jAj+ jBjcounts all the elements in only A once, and all the elements in only B once. BUT, it counts the elements in A and B TWICE.

    • [PDF File]Section 7.2: Venn Diagrams and Cardinality

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      The cardinality of A ⋂ B is 3, since A ⋂ B = {2, 4, 6}, which contains 3 elements. Example 5 What is the cardinality of P = the set of English names for the months of the year? The cardinality of this set is 12, since there are 12 months in the year. Sometimes we may be interested in the cardinality of the union or intersection of sets, but not

    • [PDF File]Chapter 7 Cardinality of sets - Web hosting

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      4 CHAPTER 7. CARDINALITY OF SETS Corollary 7.2.1 suggests a way that we can start to measure the \size" of in nite sets. We will say that any sets A and B have the same cardinality, and write jAj= jBj, if A and B can be put into 1-1 correspondence. If A can be put into 1-1 correspondence with a subset of B (that is, there is a 1-1

    • [PDF File]Basic Concepts of Set Theory, Functions and Relations

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      The number of elements in a set A is called the cardinality of A, written A . The cardinality of a finite set is a natural number. Infinite sets also have cardinalities but they are not natural numbers. We will discuss cardinal ities of infinite sets a little later (Chapter 4). 2 Be careful about “if and only if”; its abbreviation is iff.

    • [PDF File]Discrete Mathematics & Mathematical Reasoning Cardinality

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      Cardinality of sets Definition Two sets A and B have the same cardinality, jAj= jBj, iff there exists a bijection from A to B jAj jBjiff there exists an injection from A to B jAj< jBjiff jAj jBjand jAj6= jBj(A smaller cardinality than B) Unlike finite sets, for infinite sets A ˆB and jAj= jBj Even = f2n jn 2NgˆN and jEvenj= jNj

    • [PDF File]Sets and set operations

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      – Sets = collection of objects Examples of discrete structures built with the help of sets: ... Cardinality Definition: Let S be a set. If there are exactly n distinct elements in S, where n is a nonnegative integer, we say S is a finite set and that n is the cardinality of S. The cardinality of S is

    • [PDF File]Basic Structures: Sets, Functions, Sequences, Sums, and ...

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      Cardinality of Sets! ... Examples:! 1. The set of all computer science majors at your school is a subset of all students at your school.! 2. The set of integers with squares less than 100 is not a subset of the set of nonnegative integers. Another look at Equality of Sets

    • [PDF File]Chapter VIII Cardinality

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      Thus, for instance, the sets {a,b,c }and {1,2,3}have the same cardinality, which is 3. For infinite sets we cannot define the cardinality to be the number of elements, because such sets do not have any (finite) number of elements. However, there is a reason we do not just define the cardinality of an infinite set

    • [PDF File]Lecture 5: Infinities Ch 4.1 Equivalent sets and cardinality

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      Ch 4.2. Denumerable sets and the cardinal Aleph-null (ℵ0 ) We can associate with each finite set a natural number which represents its cardinality. Sets with the same cardinality form an equivalence class. Infinite sets can also be grouped into equivalence classes, such that all the sets in a given equivalence class have the same cardinality.

    • [PDF File]Cardinality of a Set

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      Cardinality of a Set We use three di erent notations for the number of elements in a nite set: I n(A) ... of the two sets, we will have counted the elements in the intersection twice. ... Examples For example, if we shu e a deck of cards and, one at a time, ...

    • [PDF File]Cardinality of infinite sets

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      Cardinality of infinite sets The cardinality |A| of a finite set A is simply the number of elements in it. When it comes to infinite sets, we no longer can speak of the number of elements in such a set. We can, however, try to match up the elements of two infinite sets A and B one by one.

    • [PDF File]Cardinality

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      cardinality as ℕ. The cardinality of the denumerable sets is denoted ℵ 0 which is read as "aleph naught" or "aleph null". (ℵ is the first letter of the Hebrew alphabet.) One may be tempted to say, in analogy with finite sets, that all denumerable sets have the same number of elements, or all denumerable sets have ℵ 0 elements.

    • [PDF File]Functions and Cardinality of Sets

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      cardinality is denoted by @ 0 (aleph-naught) and we write jAj= @ 0. A set whose cardinality is n for some natural number n is called nite. A set which is not nite is called in nite. A set of cardinality n or @ 0 is called countable; otherwise uncountable or non-denumerable. Examples. The sets N, Z, Q of natural numbers, integers, and ratio-nal ...

    • [PDF File]Math 127: Finite Cardinality

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      Before we start developing theorems, let’s get some examples working with the de nition of nite sets. Example 1. Fix m 2N. Let X m = fq 2Q j0 q 1; and mq 2Zg. Prove that X is nite, and determine its cardinality. Solution. To prove that X m is nite, by de nition we need a natural number n chosen so that we can construct a bijection from [n] to ...

    • [PDF File]CHAPTER 13 CardinalityofSets

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      SetswithEqualCardinalities 219 N because Z has all the negative integers as well as the positive ones. Definition13.1settlestheissue. Becausethebijection f :N!Z matches up Nwith Z,itfollowsthat jj˘j.Wesummarizethiswithatheorem. Theorem13.1 Thereexistsabijection f :N!Z.Therefore jNj˘jZ. The fact that N and Z have the same cardinality might prompt us ...

    • [PDF File]Discrete Maths: Exercises and Solutions

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      Sets are used extensively in counting problems, and for such applications we need to discuss the sizes of sets. Definition: Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|.

    • [PDF File]Lecture 3: Cardinality and Countability

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      countable sets { is countable. Remark: For two nite sets Aand B;we know that if Ais a strict subset of B;then Bhas cardinality greater than that of A:As the above examples show, this is not true for in nite sets. Indeed, N is a strict subset of Q;but N and Q are equicardinal! 4.

    • [PDF File]Cardinality of Sets - Gordon College

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      Sets with Equal Cardinality De nition Two sets A and B have the same cardinality, written jAj= jBj, if there exists a bijective function f : A !B. If no such bijective function exists, then the sets have unequal cardinalities, that is, jAj6= jBj. Another way to say this is that jAj= jBjif there is a one-to-one

    • [PDF File]CSC343--IntroductiontoDatabases ER Cardinality Examples ...

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      •Double (or thick) line indicates min cardinality of 1 •aka participation constraint TheEntity-RelationshipModel-- 7 CSC343--IntroductiontoDatabases An Entity Hierarchy isA isA isA isA isA TheEntity-RelationshipModel-- 8 CSC343--IntroductiontoDatabases Used when we have to model a relationship involving (entity sets and) and a relationship set.

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