CS 40: Examination - UCSB Computer Science
... 5. (12 points) Give an explicit formula for a function from the set of integers to the set of positive integers that is: (a) one-to-one, but not onto f (x) = 2x + 2, when x ≥ 0, and f (x) = −2x + 1, when x < 0. We do not need to express it as a 1-part formula, but we can: f (x) = 2|x| + 1 + b2x c/2 ...
... 5. (12 points) Give an explicit formula for a function from the set of integers to the set of positive integers that is: (a) one-to-one, but not onto f (x) = 2x + 2, when x ≥ 0, and f (x) = −2x + 1, when x < 0. We do not need to express it as a 1-part formula, but we can: f (x) = 2|x| + 1 + b2x c/2 ...
The r-Bell Numbers
... If we construct partitions on n + r elements and the first r elements are in distinct blocks, then we have two possibilities: 1) the last element, n + r, belongs to a block containing one of the first elements. Such partition can be constructed such that we construct a partition of {1, 2, . . . , n ...
... If we construct partitions on n + r elements and the first r elements are in distinct blocks, then we have two possibilities: 1) the last element, n + r, belongs to a block containing one of the first elements. Such partition can be constructed such that we construct a partition of {1, 2, . . . , n ...
Chapter 4 – Formulas and Negative Numbers
... again $5. How much has the stock price fallen overall? These are the types of problems we will be looking at in this section. When the stock decreases, that corresponds to a negative value. When the stock increases, that corresponds to a positive value. How do we add these quantities? Let’s review. ...
... again $5. How much has the stock price fallen overall? These are the types of problems we will be looking at in this section. When the stock decreases, that corresponds to a negative value. When the stock increases, that corresponds to a positive value. How do we add these quantities? Let’s review. ...
Properties of Real Numbers
... Properties of Addition & Multiplication: For all real #’s, a, b, c… ...
... Properties of Addition & Multiplication: For all real #’s, a, b, c… ...
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... There are 25 problems on this exam. Obviously, you can’t spend the same amount of time on each problem as on the midterm exams. The instructions have been modified to reflect this, so please follow them carefully. Your strategy is to do as many of the problems as you can, starting with the easiest o ...
... There are 25 problems on this exam. Obviously, you can’t spend the same amount of time on each problem as on the midterm exams. The instructions have been modified to reflect this, so please follow them carefully. Your strategy is to do as many of the problems as you can, starting with the easiest o ...
HERE
... A conservation-of-area argument shows that a square of dimension (10 + 5) by (10 + 5) yields four rectangular areas: one that is 10 by 10, one that is 5 by 5, and two that are 10 by 5. This model does not reveal why area is relevant to a discussion of multiplication, so one should begin by defining ...
... A conservation-of-area argument shows that a square of dimension (10 + 5) by (10 + 5) yields four rectangular areas: one that is 10 by 10, one that is 5 by 5, and two that are 10 by 5. This model does not reveal why area is relevant to a discussion of multiplication, so one should begin by defining ...
Vectors and Vector Operations
... vectors is sometimes regarded as a secondary operation because it can be expressed in terms of addition and negation, i.e. ...
... vectors is sometimes regarded as a secondary operation because it can be expressed in terms of addition and negation, i.e. ...
Junior Individual Test
... 4. One afternoon Marija notices that the current time is 10% of the way from 3:00 to 4:00. What fraction (in lowest terms) of the time has elapsed from 2:00 to 5:00? 5. Find the largest integer n for which 12n evenly divides 20! 6. A man is climbing a 75 foot cliff. He climbs up 12 feet every 8 minu ...
... 4. One afternoon Marija notices that the current time is 10% of the way from 3:00 to 4:00. What fraction (in lowest terms) of the time has elapsed from 2:00 to 5:00? 5. Find the largest integer n for which 12n evenly divides 20! 6. A man is climbing a 75 foot cliff. He climbs up 12 feet every 8 minu ...
Number Systems 2
... To determine this, we must get an idea of what the graph looks like and see if it passes the vertical line test So, on eth right, if we look at the graph, we clearly see it passes the vertical line test and indeed is a function!! ...
... To determine this, we must get an idea of what the graph looks like and see if it passes the vertical line test So, on eth right, if we look at the graph, we clearly see it passes the vertical line test and indeed is a function!! ...
Lesson 2
... We have already talked about plotting integers on the number line. It gives a visual representation of which number is bigger, smaller, etc. It would therefore be helpful to plot non-integer rational numbers (fractions) on the number line also. There are 2 ways to graph rational numbers on the numbe ...
... We have already talked about plotting integers on the number line. It gives a visual representation of which number is bigger, smaller, etc. It would therefore be helpful to plot non-integer rational numbers (fractions) on the number line also. There are 2 ways to graph rational numbers on the numbe ...
Addition
Addition (often signified by the plus symbol ""+"") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together; making a total of 5 apples. This observation is equivalent to the mathematical expression ""3 + 2 = 5"" i.e., ""3 add 2 is equal to 5"".Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers, rational numbers, real numbers and complex numbers and other abstract objects such as vectors and matrices.In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly.Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some non-human animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.