The Foundations: Logic and Proofs
... Introduction • Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. – Important for counting. – Programming languages have set operations. • Set theory is an important branch of mathematics. – Many different systems of axioms have been used to devel ...
... Introduction • Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. – Important for counting. – Programming languages have set operations. • Set theory is an important branch of mathematics. – Many different systems of axioms have been used to devel ...
5.1 Divisibility
... • 30,385 is divisible by 5 since it ends in 5 or 0 • 593,777,832 is divisible by 8 since the 8|832 • 543,186 is divisible by 9 since 5 + 4 + 3 + 1 + 8 + 6= 27 is divisible by 9 • 35,780 is divisible by 10 since it ends in 0 • 614,608,176 is divisible by 12 since 3 and 4 divide it ...
... • 30,385 is divisible by 5 since it ends in 5 or 0 • 593,777,832 is divisible by 8 since the 8|832 • 543,186 is divisible by 9 since 5 + 4 + 3 + 1 + 8 + 6= 27 is divisible by 9 • 35,780 is divisible by 10 since it ends in 0 • 614,608,176 is divisible by 12 since 3 and 4 divide it ...
Positive Integers
... Here we can see that the GCF(36, 84) = 12. However, listing factors can become very time consuming, especially in the case of larger numbers. Question: Is there a more efficient method for finding the GCF? Answer: Yes! ...
... Here we can see that the GCF(36, 84) = 12. However, listing factors can become very time consuming, especially in the case of larger numbers. Question: Is there a more efficient method for finding the GCF? Answer: Yes! ...
A Subrecursive Refinement of the Fundamental Theorem of Algebra
... Γ1 (f¯), Γ2 (f¯), Γ3 (f¯), Γ4 (f¯), Γ5 (f¯), Γ6 (f¯) is a representation of the complex number ϕ(ζ1 , . . . , ζN ). The following properties are evident or easily provable.4 1. All functions ϕ from CN into C that have the form ϕ(z1 , . . . , zN ) = zj with j ∈ {1, . . . , N } are E 2 -computable. 2. ...
... Γ1 (f¯), Γ2 (f¯), Γ3 (f¯), Γ4 (f¯), Γ5 (f¯), Γ6 (f¯) is a representation of the complex number ϕ(ζ1 , . . . , ζN ). The following properties are evident or easily provable.4 1. All functions ϕ from CN into C that have the form ϕ(z1 , . . . , zN ) = zj with j ∈ {1, . . . , N } are E 2 -computable. 2. ...
Full text
... That reflections of light rays within two glass plates can be expressed in terms of the Fibonacci numbers is well known [Moser, 1], In fact* if one starts with a single light ray and if the surfaces of the glass plates are half-mirrors such that they both transmit and reflect light, the number of po ...
... That reflections of light rays within two glass plates can be expressed in terms of the Fibonacci numbers is well known [Moser, 1], In fact* if one starts with a single light ray and if the surfaces of the glass plates are half-mirrors such that they both transmit and reflect light, the number of po ...
The Inclusion Exclusion Principle
... various values. We can include the 3 or not, include the 6 or not, include 0, 1, or 2 of the 4’s, 0,1, or 2 of the 5’s, and 0,1,2, or 3 of the 7’s. This number is ...
... various values. We can include the 3 or not, include the 6 or not, include 0, 1, or 2 of the 4’s, 0,1, or 2 of the 5’s, and 0,1,2, or 3 of the 7’s. This number is ...
UNC Charlotte 2009 Comprehensive
... Solution: B. First, suppose that there is exactly one pair of opposite faces of the same color. In this case, there is just one cube with the blue faces opposite, one with the white faces opposite, and one with the red faces opposite. There are no cubes with exactly two pairs of opposite faces of th ...
... Solution: B. First, suppose that there is exactly one pair of opposite faces of the same color. In this case, there is just one cube with the blue faces opposite, one with the white faces opposite, and one with the red faces opposite. There are no cubes with exactly two pairs of opposite faces of th ...
8th Grade
... 15. A troll and an ogre take jobs at the monster factory. The troll is very greedy and asks for $100 for his first day’s work, $200 for his second day’s work, $300 for his third day’s work, and so on. (Each day he makes $100 more than he did the previous day.) The ogre says he does not care much for ...
... 15. A troll and an ogre take jobs at the monster factory. The troll is very greedy and asks for $100 for his first day’s work, $200 for his second day’s work, $300 for his third day’s work, and so on. (Each day he makes $100 more than he did the previous day.) The ogre says he does not care much for ...
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.