Sets and Operations on Sets
... A Venn Diagram is often used to show relationships between sets. At right is a Venn Diagram for two sets A and B. Anything inside the circle labeled A is considered part of set A and similarly for B. We shade in the section of the diagram we are interested in, so in the figure at left, we have shade ...
... A Venn Diagram is often used to show relationships between sets. At right is a Venn Diagram for two sets A and B. Anything inside the circle labeled A is considered part of set A and similarly for B. We shade in the section of the diagram we are interested in, so in the figure at left, we have shade ...
lecture notes 4
... Prove that some two adjacent squares (sharing a side) contain numbers differing by at least 5. In the first problem, we can probably assume that residues modulo n will be the pigeons. With this in mind, we should need only about n pigeons. Given two subsets whose elements give equivalent remainders, ...
... Prove that some two adjacent squares (sharing a side) contain numbers differing by at least 5. In the first problem, we can probably assume that residues modulo n will be the pigeons. With this in mind, we should need only about n pigeons. Given two subsets whose elements give equivalent remainders, ...
An ordered partition of a set is a sequence of pairwise disjoint
... Because a string looks the same no matter which of the 3 A’s, for example, is placed first, we count the number of distinguishable strings by regarding this as an ordered partition: An unordered partition results when the collection of pairwise disjoint nonempty subsets for which the union of these ...
... Because a string looks the same no matter which of the 3 A’s, for example, is placed first, we count the number of distinguishable strings by regarding this as an ordered partition: An unordered partition results when the collection of pairwise disjoint nonempty subsets for which the union of these ...
35th IMO 1994 A1. Let m and n be positive integers. Let a 1,a2,...,am
... on the line AB and F lies on the line AC such that E, Q, F are distinct and collinear Prove that OQ is perpendicular to EF iff QE = QF . A3. For any positive integer k, let f (k) be the number of elements in the set {k + 1, k + 2, . . . , 2k} which have exactly three 1s when written in base 2. Prove ...
... on the line AB and F lies on the line AC such that E, Q, F are distinct and collinear Prove that OQ is perpendicular to EF iff QE = QF . A3. For any positive integer k, let f (k) be the number of elements in the set {k + 1, k + 2, . . . , 2k} which have exactly three 1s when written in base 2. Prove ...
Math 150 Exam 1 October 4, 2006 Choose 7 from the following 9
... 6.) Use the pigeonhole principle to prove that in a group of n people where n > 1, there are at least 2 people who have the same number of acquaintances. State where you use the pigeonhole principle. Number the people 1 through n. We will assume that all acquaintances are mutual. We will also assume ...
... 6.) Use the pigeonhole principle to prove that in a group of n people where n > 1, there are at least 2 people who have the same number of acquaintances. State where you use the pigeonhole principle. Number the people 1 through n. We will assume that all acquaintances are mutual. We will also assume ...
File
... An element is written just one time even if it exists in both of the sets Union of the two sets is commutative If A and B are two sets, then A ∪ B = B ∪ A Union of sets is also associative If A, B and C are three sets, then A ∪ (B ∪ C) = (A ∪ B) ∪ C ...
... An element is written just one time even if it exists in both of the sets Union of the two sets is commutative If A and B are two sets, then A ∪ B = B ∪ A Union of sets is also associative If A, B and C are three sets, then A ∪ (B ∪ C) = (A ∪ B) ∪ C ...
MATH 327 - Winona State University
... Course Title: Foundations of Mathematics Number of Credits: 4 Catalog Description: With an emphasis on mathematical proof writing, the following topics are covered: structure of the real and complex numbers, elementary number theory, introductory group and field properties, basic topology of the rea ...
... Course Title: Foundations of Mathematics Number of Credits: 4 Catalog Description: With an emphasis on mathematical proof writing, the following topics are covered: structure of the real and complex numbers, elementary number theory, introductory group and field properties, basic topology of the rea ...
Exercises about Sets
... a) Write all of the subset relations that exist between A, B, C, and D. b) Compute A B. Draw a Venn diagram to illustrate. c) Compute A B Draw a Venn diagram to illustrate. d) Compute B C D Draw a Venn diagram to illustrate. e) Compute B C D Draw a Venn diagram to illustrate. f) Compute B ...
... a) Write all of the subset relations that exist between A, B, C, and D. b) Compute A B. Draw a Venn diagram to illustrate. c) Compute A B Draw a Venn diagram to illustrate. d) Compute B C D Draw a Venn diagram to illustrate. e) Compute B C D Draw a Venn diagram to illustrate. f) Compute B ...
Lecture Notes 2: Infinity
... Subset of a set S: A set that has some of the members of S. Proper subset: subset that does not have all the members of S. For finite sets, “the whole is greater than the part.” An infinite set can have proper subsets that are in 1-to-1 correspondence with the whole set. (“Hilbert’s Hotel”) ...
... Subset of a set S: A set that has some of the members of S. Proper subset: subset that does not have all the members of S. For finite sets, “the whole is greater than the part.” An infinite set can have proper subsets that are in 1-to-1 correspondence with the whole set. (“Hilbert’s Hotel”) ...
Chapter 4 Set Theory
... justify than what we had intuitively before: both sets are equal because whenever a number belongs to one, it belongs to the other. Definition 28. The cardinality of a set S is the number of distinct elements of S. If |S| is finite, the set is said to be finite. It is said to be infinite otherwise. ...
... justify than what we had intuitively before: both sets are equal because whenever a number belongs to one, it belongs to the other. Definition 28. The cardinality of a set S is the number of distinct elements of S. If |S| is finite, the set is said to be finite. It is said to be infinite otherwise. ...
