Section 6.5: Combinations Example Recall our five friends, Alan
... beginning of the previous section. The have won 3 tickets for a concert in Chicago and everybody would like to go. However, they cannot afford two more tickets and must choose a group of three people from the five to go to the concert. Alan decides to make a fair decision on who gets the tickets, by ...
... beginning of the previous section. The have won 3 tickets for a concert in Chicago and everybody would like to go. However, they cannot afford two more tickets and must choose a group of three people from the five to go to the concert. Alan decides to make a fair decision on who gets the tickets, by ...
2.1 Function/Relation Notes
... 3. Using a verbal description to identify a function a. Consider the relationship of CHS student ID numbers and student locker numbers. Write “ID #” in the input box and “locker number” in the output box. Is a student’s locker number a function of their ID number? Why or why not? Complete the boxes ...
... 3. Using a verbal description to identify a function a. Consider the relationship of CHS student ID numbers and student locker numbers. Write “ID #” in the input box and “locker number” in the output box. Is a student’s locker number a function of their ID number? Why or why not? Complete the boxes ...
MATH 311–01 Exam #1 Solutions 1. (7 points) Consider the true
... is that nn = n2 , which contributes only a single factor to the list. So when we write out, for instance, all the ways to write 36 as a product of two numbers we have 1 · 36, 2 · 18, 3 · 12, 4 · 9, and 6 · 6. The first four products describe two factors each, while the last describes only one—and be ...
... is that nn = n2 , which contributes only a single factor to the list. So when we write out, for instance, all the ways to write 36 as a product of two numbers we have 1 · 36, 2 · 18, 3 · 12, 4 · 9, and 6 · 6. The first four products describe two factors each, while the last describes only one—and be ...
Lecture 9 - CSE@IIT Delhi
... Question: In a party of n people, is it always true that there are two people shaking hands with the same number of people? Everyone can shake hand with 0 to n-1 people, and there are n people, and so it does not seem that it must be the case, but think about it carefully: Case 1: if there is a pers ...
... Question: In a party of n people, is it always true that there are two people shaking hands with the same number of people? Everyone can shake hand with 0 to n-1 people, and there are n people, and so it does not seem that it must be the case, but think about it carefully: Case 1: if there is a pers ...
Lecture 10: Combinatorics 1 Binomial coefficient and Pascals triangle
... How many partitions of [n] in k blocks exists? These numbers are known as Stirling numbers of the second kind. Definition 4.2 The Stirling numbers of the second kind, S(n, k), describe the number of possible ways to partition a set of n elements into k blocks. S(n, 0) = 0 if n > 0 S(n, 1) = 1 S(n, n ...
... How many partitions of [n] in k blocks exists? These numbers are known as Stirling numbers of the second kind. Definition 4.2 The Stirling numbers of the second kind, S(n, k), describe the number of possible ways to partition a set of n elements into k blocks. S(n, 0) = 0 if n > 0 S(n, 1) = 1 S(n, n ...
Real Numbers - Will Rosenbaum
... field (as is R), but the complex numbers C are not an ordered field. Problem 6. Prove that C is not an ordered field. Specifically, there is no order relation < on C which satisfies all of the order axioms. The ordering of R is what allows us to visualize R as the “real number line.” Smaller numbers ...
... field (as is R), but the complex numbers C are not an ordered field. Problem 6. Prove that C is not an ordered field. Specifically, there is no order relation < on C which satisfies all of the order axioms. The ordering of R is what allows us to visualize R as the “real number line.” Smaller numbers ...
answers
... F , then for every n = 1, 2, ..., there would exist a point yn ∈ F such that yn ∈ B(x, n1 ). But then this would describe a sequence yn which converges to x ∈ F c , an impossibility. Conclude that x is not a limit point of F for all x ∈ F c , and thus that F is closed. Now suppose that F is closed. ...
... F , then for every n = 1, 2, ..., there would exist a point yn ∈ F such that yn ∈ B(x, n1 ). But then this would describe a sequence yn which converges to x ∈ F c , an impossibility. Conclude that x is not a limit point of F for all x ∈ F c , and thus that F is closed. Now suppose that F is closed. ...
Cardinality, countable and uncountable sets
... looks very much like “an equivalence relation in the class of all sets”, and indeed this can be formalized in axiomatic set theory, but we’ll leave that for the advanced course. The notion of “cardinality” of a set was develop in the late 19th/early 20th centuries by the German mathematician Georg C ...
... looks very much like “an equivalence relation in the class of all sets”, and indeed this can be formalized in axiomatic set theory, but we’ll leave that for the advanced course. The notion of “cardinality” of a set was develop in the late 19th/early 20th centuries by the German mathematician Georg C ...
1 Sets
... set {a, b, c}; of course it is also a multiset over the set {a, b, c, d} with 0 copies of d. Let M be a multiset of n objects over S. If S has k elements and is ordered as S = {x1 , x2 , . . . , xk }, we say that a multiset M over S if of type (r1 , r2 , . . . , rk ) if M has r1 copies of the 1st ob ...
... set {a, b, c}; of course it is also a multiset over the set {a, b, c, d} with 0 copies of d. Let M be a multiset of n objects over S. If S has k elements and is ordered as S = {x1 , x2 , . . . , xk }, we say that a multiset M over S if of type (r1 , r2 , . . . , rk ) if M has r1 copies of the 1st ob ...
UProperty 1
... The problem then is, what else is there on the number line? But what are these irrationals (at the moment, we just know that they are not rational numbers). In other words, what exactly is the set of real numbers ? There have been many attempts to define this set rigorously. All the study of calcul ...
... The problem then is, what else is there on the number line? But what are these irrationals (at the moment, we just know that they are not rational numbers). In other words, what exactly is the set of real numbers ? There have been many attempts to define this set rigorously. All the study of calcul ...
Math 554 - Fall 08 Lecture Note Set # 1
... Defn. Suppose that S is an ordered set and A ⊆ S. An element β ∈ S is said to be an upper bound for A if a ≤ β, ∀a ∈ A. An element α is said to be a least upper bound for A if 1. α is an upper bound for A 2. if β is any upper bound for A, then α ≤ β. In this case, the supremum of A (=: sup A) is def ...
... Defn. Suppose that S is an ordered set and A ⊆ S. An element β ∈ S is said to be an upper bound for A if a ≤ β, ∀a ∈ A. An element α is said to be a least upper bound for A if 1. α is an upper bound for A 2. if β is any upper bound for A, then α ≤ β. In this case, the supremum of A (=: sup A) is def ...
