Mathematical Reasoning: Writing and Proof

... Following are some of the important features of this text that will help with the transition from calculus to upper-level mathematics courses. 1. Emphasis on Writing in Mathematics Issues dealing with writing mathematical exposition are addressed throughout the book. Guidelines for writing mathemati ...

... Following are some of the important features of this text that will help with the transition from calculus to upper-level mathematics courses. 1. Emphasis on Writing in Mathematics Issues dealing with writing mathematical exposition are addressed throughout the book. Guidelines for writing mathemati ...

... Following are some of the important features of this text that will help with the transition from calculus to upper-level mathematics courses. 1. Emphasis on Writing in Mathematics Issues dealing with writing mathematical exposition are addressed throughout the book. Guidelines for writing mathemati ...

Untitled

... systems, particularly for the discussion of the cardinality of sets in Chapter 6, but it was always somewhat out of place given the level and scope of this text. The background material needed for Chapter 6 has now been summarized in a new section at the start of that chapter, making the chapter bot ...

... systems, particularly for the discussion of the cardinality of sets in Chapter 6, but it was always somewhat out of place given the level and scope of this text. The background material needed for Chapter 6 has now been summarized in a new section at the start of that chapter, making the chapter bot ...

A Transition to Advanced Mathematics

... permutations of a set. In Section 5.3 on countable sets, the major results (that subsets and unions of countably many countable sets are countable) are moved up to make them more accessible. In Chapter 7, there is even more emphasis on the meaning of the completeness property of the real number syst ...

... permutations of a set. In Section 5.3 on countable sets, the major results (that subsets and unions of countably many countable sets are countable) are moved up to make them more accessible. In Chapter 7, there is even more emphasis on the meaning of the completeness property of the real number syst ...

MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set

... c(A ∪ B) = c(A) + c(B) − c(A ∩ B) Example Each of the next examples leads to another useful counting rule. (a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(A ∩ C ). (b) Let N = {0, 1, 2, . . .} be the set of natural numbers. Find c(N). (c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(D ...

... c(A ∪ B) = c(A) + c(B) − c(A ∩ B) Example Each of the next examples leads to another useful counting rule. (a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(A ∩ C ). (b) Let N = {0, 1, 2, . . .} be the set of natural numbers. Find c(N). (c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(D ...

Book of Proof

... ©ª ©ª empty set is the set that has no elements. We denote it as ;, so ; = . ©ª Whenever you see the symbol ;, it stands for . Observe that |;| = 0. The empty set is the only set whose cardinality is zero. © ª Be very careful how you write the empty set. Don’t write ; when you mean ;. These sets can ...

... ©ª ©ª empty set is the set that has no elements. We denote it as ;, so ; = . ©ª Whenever you see the symbol ;, it stands for . Observe that |;| = 0. The empty set is the only set whose cardinality is zero. © ª Be very careful how you write the empty set. Don’t write ; when you mean ;. These sets can ...

Fibonacci pitch sets

... 2.3 Prime Sets and Subsets The absence of certain residue classes in f1 and f5 is a direct result of the modulus used; modulo 12 is said to be a defective modulus,7 as the Fibonacci sequence doesn’t contain a complete system of residues modulo 12.8 Residue class 6 is absent in f1. This results from ...

... 2.3 Prime Sets and Subsets The absence of certain residue classes in f1 and f5 is a direct result of the modulus used; modulo 12 is said to be a defective modulus,7 as the Fibonacci sequence doesn’t contain a complete system of residues modulo 12.8 Residue class 6 is absent in f1. This results from ...

Section 3.3 Equivalence Relation

... Classifying objects and placing similar objects into groups provides a way to organize information and focus attention on the similarities of like objects and not on the dissimilarities of dislike objects. Mathematicians have been classifying objects for millennia. Lines in the plane can be subdivid ...

... Classifying objects and placing similar objects into groups provides a way to organize information and focus attention on the similarities of like objects and not on the dissimilarities of dislike objects. Mathematicians have been classifying objects for millennia. Lines in the plane can be subdivid ...

(it), sem. -iii, logic and discrete mathematics

... For example, collection of fans in a class room collection of all people in a state etc. Now, consider the example, collection of brave people in a class. Is it a set? The answer is no because brave is a relative word and it varies from person to person so it is not a set. Note : Well-defined means ...

... For example, collection of fans in a class room collection of all people in a state etc. Now, consider the example, collection of brave people in a class. Is it a set? The answer is no because brave is a relative word and it varies from person to person so it is not a set. Note : Well-defined means ...

Lecture Notes for College Discrete Mathematics Szabolcs Tengely

... • Sets given by enumeration. If we have a set containing certain elements, then we enclose these elements in braces. For example, if is a set containing 1, 2 and 3 we write . This notation is difficult to use if the given set has large amount of elements. In this case we list only some (usually cons ...

... • Sets given by enumeration. If we have a set containing certain elements, then we enclose these elements in braces. For example, if is a set containing 1, 2 and 3 we write . This notation is difficult to use if the given set has large amount of elements. In this case we list only some (usually cons ...

Comparing sizes of sets

... Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A to B iff |A| = |B|. Let’s do it. . . Take arbitrary finite sets A and B. LR: Assume f : A → B is bijective. Then f is inje ...

... Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A to B iff |A| = |B|. Let’s do it. . . Take arbitrary finite sets A and B. LR: Assume f : A → B is bijective. Then f is inje ...

1 - CS285

... Imagine a set S containing n elements and a set T containing (n + 1) elements, namely all elements in S plus a new element a. Calculating C(n + 1, k) is equivalent to answering the question: How many subsets of T containing k items are there? Case I: The subset contains (k – 1) elements of S plus th ...

... Imagine a set S containing n elements and a set T containing (n + 1) elements, namely all elements in S plus a new element a. Calculating C(n + 1, k) is equivalent to answering the question: How many subsets of T containing k items are there? Case I: The subset contains (k – 1) elements of S plus th ...