Lecture 2 – Proof Techniques
... is not direct which don’t start with the hypothesis and end with the conclusion ( we call it indirect proof) Indirect proof (proof by contraposition): Assume q, and prove p. Contraposition (pq q p) We take q as hypothesis , and using axioms , definitions any proven theorem to follow p ...
... is not direct which don’t start with the hypothesis and end with the conclusion ( we call it indirect proof) Indirect proof (proof by contraposition): Assume q, and prove p. Contraposition (pq q p) We take q as hypothesis , and using axioms , definitions any proven theorem to follow p ...
Po Leung Kuk 15 Primary Mathematics World Contest Team Contest 2012
... give them to pile B. Step 2: From pile B, take the same number of balls as in the current pile A and give them to pile A. Step 3: From pile A, take the same number of balls as in the current pile B and give them to pile B. Step 4: From pile B, take the same number of balls as in the current pile A a ...
... give them to pile B. Step 2: From pile B, take the same number of balls as in the current pile A and give them to pile A. Step 3: From pile A, take the same number of balls as in the current pile B and give them to pile B. Step 4: From pile B, take the same number of balls as in the current pile A a ...
Y8 Spring Term Units Document
... A. Read and write positive integer powers of 10; multiply and divide integers and decimals by 0.1, 0.01. B. Order decimals. C. Round positive numbers to any given power of 10; round decimals to the nearest whole number or to one or two decimal places. D. Consolidate and extend mental methods of calc ...
... A. Read and write positive integer powers of 10; multiply and divide integers and decimals by 0.1, 0.01. B. Order decimals. C. Round positive numbers to any given power of 10; round decimals to the nearest whole number or to one or two decimal places. D. Consolidate and extend mental methods of calc ...
Single Digits: In Praise of Small Numbers
... set that can be “listed” has the same cardinality as the positive integers. Such sets are called countably infinite, or countable for short. How about comparing the set of positive integers to the set of positive rational numbers? Cantor claimed that these two sets have the same size. Again, how can ...
... set that can be “listed” has the same cardinality as the positive integers. Such sets are called countably infinite, or countable for short. How about comparing the set of positive integers to the set of positive rational numbers? Cantor claimed that these two sets have the same size. Again, how can ...
here
... We first analyse the set of positive integers n > 5 for which n − a2 is four times a prime number for any positive odd integer a such that a2 ≤ n and for which n − a2 is a prime number for any positive even integer a such that a2 ≤ n. There are only three numbers with these properties: n = 21, 77, 4 ...
... We first analyse the set of positive integers n > 5 for which n − a2 is four times a prime number for any positive odd integer a such that a2 ≤ n and for which n − a2 is a prime number for any positive even integer a such that a2 ≤ n. There are only three numbers with these properties: n = 21, 77, 4 ...
Lec11Proofs
... Lets say we want to prove pq but the premise p can be shown to be false! Then pq is always true because (FT) = T and (FF) = T. This is a vacuous prove. Old example: prove that for any set S: S Proof: The following must be shown to be true: x ( x However: the empty set does not contain a ...
... Lets say we want to prove pq but the premise p can be shown to be false! Then pq is always true because (FT) = T and (FF) = T. This is a vacuous prove. Old example: prove that for any set S: S Proof: The following must be shown to be true: x ( x However: the empty set does not contain a ...
PDF Version of module - Australian Mathematical Sciences Institute
... No mathematician has been able to establish whether or not this conjecture is true, making it one of the world’s longest-standing unsolved mathematical problems. A prime pair consists of two primes with a difference of 2, like 3 & 5, 11 & 13, 71 & 73. Prime pairs stand out in the list of primes up t ...
... No mathematician has been able to establish whether or not this conjecture is true, making it one of the world’s longest-standing unsolved mathematical problems. A prime pair consists of two primes with a difference of 2, like 3 & 5, 11 & 13, 71 & 73. Prime pairs stand out in the list of primes up t ...
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.