Recurrence relations and generation functions
... Determine the number of ways to color the squares of a 1-by-n chessboard, usign the colors red, white, and blue, if an even number of squares are to be colored red. Hints: Let hn denote the number of such colorings where we define h0 =1. Then hn equals tgeh number of n-permutations of a multiset of ...
... Determine the number of ways to color the squares of a 1-by-n chessboard, usign the colors red, white, and blue, if an even number of squares are to be colored red. Hints: Let hn denote the number of such colorings where we define h0 =1. Then hn equals tgeh number of n-permutations of a multiset of ...
Cardinality: Counting the Size of Sets ()
... The conjecture that the answer to the above question is “yes” is known as the continuum hypothesis. This problem was first posed by Cantor in 1874 and became a major unsolved problem in the following decades. The question was fully answered, though, through two results, one in 1940 and the other in ...
... The conjecture that the answer to the above question is “yes” is known as the continuum hypothesis. This problem was first posed by Cantor in 1874 and became a major unsolved problem in the following decades. The question was fully answered, though, through two results, one in 1940 and the other in ...
ppt
... countable, the other is not. • Q: Is there a set whose cardinality is “inbetween”? • Q: Is the cardinality of R the same as that of [0,1) ? ...
... countable, the other is not. • Q: Is there a set whose cardinality is “inbetween”? • Q: Is the cardinality of R the same as that of [0,1) ? ...
Spring 2007 Math 510 Hints for practice problems
... children. Each child has the ability to take as many as he/she is given. This becomes problem of 10-combinations of a multiset with three types of objects (apples received by each child) with ...
... children. Each child has the ability to take as many as he/she is given. This becomes problem of 10-combinations of a multiset with three types of objects (apples received by each child) with ...
Section 2.2
... Note: Definitions 2.2.6 and 2.2.7 are not really much different from each other, but the second one has a bit more technical machinery of a sort that is sometimes useful in trying to describe correspondences. The example about the keys and rooms is a case of both. The sets A and B here are respective ...
... Note: Definitions 2.2.6 and 2.2.7 are not really much different from each other, but the second one has a bit more technical machinery of a sort that is sometimes useful in trying to describe correspondences. The example about the keys and rooms is a case of both. The sets A and B here are respective ...
polynomial function
... based on its weight, x, in pounds is given by C(x) = 0.03x 3 – 0.75x 2 + 4.5x + 7. a. What is the cost of shipping a 7-pound gift basket? b. What is the cost of shipping a 19-pound gift basket? 2. Reasoning The total number of lights in a triangular lighting rig is related to the triangular numbers, ...
... based on its weight, x, in pounds is given by C(x) = 0.03x 3 – 0.75x 2 + 4.5x + 7. a. What is the cost of shipping a 7-pound gift basket? b. What is the cost of shipping a 19-pound gift basket? 2. Reasoning The total number of lights in a triangular lighting rig is related to the triangular numbers, ...
PDF
... By means of these formulae, one may derive some important properties of the central binomial coeficients. By examining the first two formulae, one may deduce results about the prime factors of central binomial coefficients (for proofs, please see the attachments to this entry): Theorem 1 If n ≥ 3 i ...
... By means of these formulae, one may derive some important properties of the central binomial coeficients. By examining the first two formulae, one may deduce results about the prime factors of central binomial coefficients (for proofs, please see the attachments to this entry): Theorem 1 If n ≥ 3 i ...
Combining Signed Numbers
... The easiest way to describe many sets is by using words. B = {all blue-eyed blonds in this class} There isn’t a really good mathematical equivalent for that! ...
... The easiest way to describe many sets is by using words. B = {all blue-eyed blonds in this class} There isn’t a really good mathematical equivalent for that! ...
1 Sets
... that A, B, C, . . .; elements of sets are usually denoted by lowercase letters such that a, b, c, . . . . There are two ways to express a set. One is to list all elements of the set; the other one is to point out the attributes for the elements of the set. For instance, A = {1, −1}; B = {x | x real ...
... that A, B, C, . . .; elements of sets are usually denoted by lowercase letters such that a, b, c, . . . . There are two ways to express a set. One is to list all elements of the set; the other one is to point out the attributes for the elements of the set. For instance, A = {1, −1}; B = {x | x real ...
Infinity
... appears to me insoluble. Since it is clear that we may have one line segment longer than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line seg ...
... appears to me insoluble. Since it is clear that we may have one line segment longer than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line seg ...
§2.1: Basic Set Concepts MGF 1106-Peace Def: A set is a collection
... -Recall the natural numbers are all whole numbers starting at 1 and going up: {1, 2, 3, 4, …} -Since it comes up so much, we have a special name for this set: N = the set of all natural numbers. This is an infinite set, meaning it goes on forever. Ex. Represent the following set in roster method: Th ...
... -Recall the natural numbers are all whole numbers starting at 1 and going up: {1, 2, 3, 4, …} -Since it comes up so much, we have a special name for this set: N = the set of all natural numbers. This is an infinite set, meaning it goes on forever. Ex. Represent the following set in roster method: Th ...
INFINITY: CARDINAL NUMBERS 1. Some terminology of set theory
... Loosely speaking, S is smaller or equal in size to T if and only if one can match the elements of S with elements of T so that all elements of S get used (but maybe some elements of T are left over). For example, there is an injection {a, b, c} → N sending a to 1, b to 2, and c to 17; this proves th ...
... Loosely speaking, S is smaller or equal in size to T if and only if one can match the elements of S with elements of T so that all elements of S get used (but maybe some elements of T are left over). For example, there is an injection {a, b, c} → N sending a to 1, b to 2, and c to 17; this proves th ...
Document
... Definition: The cardinality of an infinite set S that is countable is denotes by א0 (where אis aleph, the first letter of the Hebrew alphabet). We write |S| = א0 and say that S has cardinality “aleph null”. Note: Georg Cantor defined the notion of cardinality and was the first to realize that ...
... Definition: The cardinality of an infinite set S that is countable is denotes by א0 (where אis aleph, the first letter of the Hebrew alphabet). We write |S| = א0 and say that S has cardinality “aleph null”. Note: Georg Cantor defined the notion of cardinality and was the first to realize that ...
Putnam Questions, Week 2 1. Prove that the number of subsets of {1
... 2. In how many ways can two squares be selected from an 8-by-8 chessboard so that they are not in the same row or the same column? 3. In how many ways can four squares, not all in the same row or column, be selected from an 8-by-8 chessboard to form a rectangle? 4. Find the number of subsets of {1, ...
... 2. In how many ways can two squares be selected from an 8-by-8 chessboard so that they are not in the same row or the same column? 3. In how many ways can four squares, not all in the same row or column, be selected from an 8-by-8 chessboard to form a rectangle? 4. Find the number of subsets of {1, ...
09-05_Travis_Hoppe_slides
... • A mathematician would call the hobo-map an undirected, unlabeled simple graph, where the process for determining two graphs are the ‘same’ is known as graph isomorphism. • Computationally, graph isomorphism is curious, it belongs to NP but it is not known to have a polynomial solution (P) nor is i ...
... • A mathematician would call the hobo-map an undirected, unlabeled simple graph, where the process for determining two graphs are the ‘same’ is known as graph isomorphism. • Computationally, graph isomorphism is curious, it belongs to NP but it is not known to have a polynomial solution (P) nor is i ...
Functions
... Image of function (in class) Inverse image of a point (in class) f=g if – f and g have the same domain – f and g have the same codomain – For all x (in domain), f(x)=g(x) ...
... Image of function (in class) Inverse image of a point (in class) f=g if – f and g have the same domain – f and g have the same codomain – For all x (in domain), f(x)=g(x) ...
Discrete Math
... 4. Three dice are thrown. Each dice can have a square with one of {1, 2, 3, 4, 5, 6} facing up. (a) How many different possibilities are there for the numbers facing up with different colored dice? (b) How many possibilities are there with two kinds of dice? (c) How many different multisets of 3 num ...
... 4. Three dice are thrown. Each dice can have a square with one of {1, 2, 3, 4, 5, 6} facing up. (a) How many different possibilities are there for the numbers facing up with different colored dice? (b) How many possibilities are there with two kinds of dice? (c) How many different multisets of 3 num ...
Lesson 2
... • The number of elements in a set is called the cardinal number, or cardinality, of the set. • L = {a,e,i,o,u} has cardinality of 5 ...
... • The number of elements in a set is called the cardinal number, or cardinality, of the set. • L = {a,e,i,o,u} has cardinality of 5 ...
