Basic Counting
... We can use the Principle of Inclusion and Exclusion to derive a formula for (n,k) and thus for the Stirling Numbers of the Second Kind. (1) Let n and k be positive integers and let S be the set of functions from [n] into [k]. That is, S={f:[n]→[k]}. For i=1,2,…,k, let Ai={functions from [n] to [k] ...
... We can use the Principle of Inclusion and Exclusion to derive a formula for (n,k) and thus for the Stirling Numbers of the Second Kind. (1) Let n and k be positive integers and let S be the set of functions from [n] into [k]. That is, S={f:[n]→[k]}. For i=1,2,…,k, let Ai={functions from [n] to [k] ...
chapter1
... Alphabet is a finite set of symbols and is denoted by . In fact, any object can be in an alphabet; from a formal point of view, an alphabet is simply a finite set of any sort. A string over an alphabet is a finite sequence of symbols from the alphabet. A string may have no symbols at all; in this c ...
... Alphabet is a finite set of symbols and is denoted by . In fact, any object can be in an alphabet; from a formal point of view, an alphabet is simply a finite set of any sort. A string over an alphabet is a finite sequence of symbols from the alphabet. A string may have no symbols at all; in this c ...
CS 381 Midterm Review
... 11. Apply a function to a set. 12. Given a function, determine if it is injective, surjective, and/or bijective. 13. Given two function definitions, find the composition of the functions. 14. Compute the ceiling and floor of any real number. 15. Write pseudocode to solve a list-based problem. 16. Gi ...
... 11. Apply a function to a set. 12. Given a function, determine if it is injective, surjective, and/or bijective. 13. Given two function definitions, find the composition of the functions. 14. Compute the ceiling and floor of any real number. 15. Write pseudocode to solve a list-based problem. 16. Gi ...
Largest Contiguous Sum
... If you can, you have a solution to x = x + y, where x is the cardinality of the set and y >= 1 is the cardinality of the stuff you removed Impossible with finite sets Possible with infinite sets Technically, an infinite set is a set where there exists a one-to-one correspondence between the set itse ...
... If you can, you have a solution to x = x + y, where x is the cardinality of the set and y >= 1 is the cardinality of the stuff you removed Impossible with finite sets Possible with infinite sets Technically, an infinite set is a set where there exists a one-to-one correspondence between the set itse ...
B - Kutztown University
... f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f. a is called the preimage of b. The range of f is the set of all images of points in A under f. We denote it by f(A). Two functions are equal whe ...
... f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f. a is called the preimage of b. The range of f is the set of all images of points in A under f. We denote it by f(A). Two functions are equal whe ...
04-function
... • Simple case: Q+ is countable. • Each element of Q+ can be represented by a/b, where a and b are relatively-prime integers and a,b>0. b ...
... • Simple case: Q+ is countable. • Each element of Q+ can be represented by a/b, where a and b are relatively-prime integers and a,b>0. b ...
![MATH 110 MIDTERM 2 FALL 2005 ANSWERS 1. [10 pts.] Suppose I](http://s1.studyres.com/store/data/017153691_1-2d4855d7e3d694f90f839ac1345a9f9c-300x300.png)
Examples of equivalence relations
... Let X be a set and let x, y, and z be elements of X. An equivalence relation, ~, on X is a relation on X such that: Reflexive Property: x is equivalent to x for all x in X. Symmetric Property: if x is equivalent to y, then y is equivalent to x. Transitive Property: if x is equivalent to y and y is e ...
... Let X be a set and let x, y, and z be elements of X. An equivalence relation, ~, on X is a relation on X such that: Reflexive Property: x is equivalent to x for all x in X. Symmetric Property: if x is equivalent to y, then y is equivalent to x. Transitive Property: if x is equivalent to y and y is e ...
An Introduction to Equivalence Relations and Partitions 1 2 3 Set X
... Let X be a set and let x, y, and z be elements of X. An equivalence relation, ~, on X is a relation on X such that: Reflexive Property: x is equivalent to x for all x in X. Symmetric Property: if x is equivalent to y, then y is equivalent to x. Transitive Property: if x is equivalent to y and y is e ...
... Let X be a set and let x, y, and z be elements of X. An equivalence relation, ~, on X is a relation on X such that: Reflexive Property: x is equivalent to x for all x in X. Symmetric Property: if x is equivalent to y, then y is equivalent to x. Transitive Property: if x is equivalent to y and y is e ...
Section 1.1 - GEOCITIES.ws
... You shall be able to write a given interval in set-builder notation. [Problems 83 – 90] ...
... You shall be able to write a given interval in set-builder notation. [Problems 83 – 90] ...
Bijective Correspondences and Countably Infinite Sets
... Two sets have the same cardinality if �and only if) it is possible to match each element of A to an element of B in such a way that every element of each set has exactly one “partner” in the other set. This might not be particularly important/interesting in the case of finite sets, since we can often ...
... Two sets have the same cardinality if �and only if) it is possible to match each element of A to an element of B in such a way that every element of each set has exactly one “partner” in the other set. This might not be particularly important/interesting in the case of finite sets, since we can often ...
Document
... One-to-one correspondence Two sets M and N are equivalent … if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. If M and N are equivalent we often say that they have they have the ...
... One-to-one correspondence Two sets M and N are equivalent … if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. If M and N are equivalent we often say that they have they have the ...
Chap4 - Real Numbers
... One-to-one correspondence Two sets M and N are equivalent … if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. If M and N are equivalent we often say that they have they have the ...
... One-to-one correspondence Two sets M and N are equivalent … if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. If M and N are equivalent we often say that they have they have the ...