Integers!!! INTEGERS!!!! Numbers can be both positive and negative. When working with negative numbers there are rules that must be followed. Think of some ways that you use negative numbers???? Visualisation Think of a greenhouse.

ByProof Methods: Part 2 Sections 3.1-3.6 More Number Theory Definitions Divisibility n is divisible by d iff kZ | n=d*k d|n is read “d divides n” where n and d are integers and d 0 (note: d|n d/n) Other ways to say it: n is a multiple of d d is a factor of n d is a divisor of n

ByChapter 2-1 Integers and Absolute Values. 0. -5. -4. -3. -2. -1. +1. +2. +3. +4. +5. Here is a number line. An integer is any number on a number line. 0. -5. -4. -3. -2. -1. +1. +2. +3. +4. +5.

ByMathematical Problems & Inquiry in Mathematics. AME Tenth Anniversary Meeting May 29 2004 A/P Peter Pang Department of Mathematics and University Scholars Programme, NUS. Four Important Concepts. Specificity Generality Specialization Generalization. D. F. 3. 7.

ByAlgorithms, Part 1 of 3. Topics Definition of an Algorithm Algorithm Examples Syntax versus Semantics. Problem Solving. Problem solving is the process of transforming the description of a problem into the solution of that problem. We use our knowledge of the problem domain (requirements).

ByEquivalence Relations. Aaron Bloomfield CS 202 Epp, section ???. Introduction. Certain combinations of relation properties are very useful We won’t have a chance to see many applications in this course In this set we will study equivalence relations

ByCircular Linked List. EENG212 Algorithms and Data Structures. Circular Linked Lists. In linear linked lists if a list is traversed (all the elements visited) an external pointer to the list must be preserved in order to be able to reference the list again.

ByProperties of the Integers: Mathematical Induction. Chapter 4. 1. Mathematical Induction 2. Harmonic, Fibonacci, Lucas Numbers 3. Prime Numbers. Chapter 4 Properties of the Integers: Mathematical Induction . 4.1 The Well-Ordering Principle: Mathematical Induction.

ByRating Scales: What the Research Says. Joe Dumas Tom Tullis UX Consultant Fidelity Investments joe.dumas99@gmail.com tom.tullis@fmr.com. The Scope of the Session. Discussion of literature about rating scales in usability methods, primarily usability testing

ByKnapsack Cipher. 0-1 knapsack problem. Given a positive integer C and a vector A=(a 1 ,...,a n ) of positive integers, find a subset of the elements of A that sum to C; that is, find a binary vector M=(m 1 ,...,m n ) such that C=AM, or . Example of 0-1 knapsack problem.

ByProblem Spaces P/NP. P/NP. Chapter 7 of the book We’ll skip around a little bit and pull in some simpler, alternate “proofs” Intractable Problems Refer to problems we cannot solve in a reasonable time on the Turing Machine/Computer

ByCounting Permutations When Indistinguishable Objects May Exist. How many rows , each one consisting of 3 A’s 1 B, and 4 C’s are there? (Here are some such rows: BACCCAAC ABCACACC CCCCAAAB Etc.) Answer: (3+1+4)! / (3!1!4!). In general:

ByGEOMETRIC SEQUENCES. These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio . 1, 2, 4, 8, 16 . . . r = 2.

ByPROGRAMMING IN HASKELL. Chapter 5 - List Comprehensions. Set Comprehensions. In mathematics, the comprehension notation can be used to construct new sets from old sets. {x 2 | x {1...5}}. The set {1,4,9,16,25} of all numbers x 2 such that x is an element of the set {1…5}.

ByIf I told you once, it must be. Recursion. Recursive Definitions. Recursion is a principle closely related to mathematical induction. In a recursive definition , an object is defined in terms of itself. We can recursively define sequences , functions and sets .

ByRoots of Real Numbers and Radical Expressions. Definition of n th Root. For any real numbers a and b and any positive integers n , if an = b , then a is the n th root of b . ** For a square root the value of n is 2. radical. index. radicand. Notation.

ByRating Scales: What the Research Says. Joe Dumas Tom Tullis UX Consultant Fidelity Investments joe.dumas99@gmail.com tom.tullis@fmr.com. The Scope of the Session. Discussion of literature about rating scales in usability methods, primarily usability testing

ByChapter 2: Sets, Functions, Sequences, and Sums. §2.1: Sets. § 2.1 – Sets. Introduction to Set Theory. A set is a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.

ByCSCI 1900 Discrete Structures. Sets and Subsets Reading: Kolman, Section 1.1. Definitions of sets. A set is any well-defined collection of objects The elements or members of a set are the objects contained in the set

ByLoops. Partial Correctness allows the code to be non-terminating . We will start by assuming that code terminates and postpone the termination topic until later. So we will deal with the situation where if the code terminates then its final state {Q} is satisfied. Some Definitions:

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