MATHEMATICAL STATEMENTS AND PROOFS In this note we
... more interesting question of how to come up with good and relevant ideas to solve the given problem. (Thus, if we compare mathematics to the game of chess, then in this note we merely aim at teaching the rules of chess, i.e. how the various pieces are allowed to move; we do not try to say very much ...
... more interesting question of how to come up with good and relevant ideas to solve the given problem. (Thus, if we compare mathematics to the game of chess, then in this note we merely aim at teaching the rules of chess, i.e. how the various pieces are allowed to move; we do not try to say very much ...
An Introduction to Proofs and the Mathematical Vernacular 1
... By focusing on how proofs work and how they are expressed we will be learning to think about mathematics as mathematicians do. This means learning the language and notation (symbols) which we use to express our reasoning precisely. But writing a proof is always preceded by finding the logical argume ...
... By focusing on how proofs work and how they are expressed we will be learning to think about mathematics as mathematicians do. This means learning the language and notation (symbols) which we use to express our reasoning precisely. But writing a proof is always preceded by finding the logical argume ...
Math 13 — An Introduction to Abstract Mathematics
... categorically say that something is true or false. In reality we can rarely be so certain. A greasy salesman in a TV advert may claim that to have proved that a certain cream makes you look younger; a defendant may be proved guilty in court; the gravitational constant is 9.81ms−2 . Ask yourself what ...
... categorically say that something is true or false. In reality we can rarely be so certain. A greasy salesman in a TV advert may claim that to have proved that a certain cream makes you look younger; a defendant may be proved guilty in court; the gravitational constant is 9.81ms−2 . Ask yourself what ...
Lecture Notes - Alistair Savage
... statement is true, we may then use it in following statements. In general, we would like to assume as few axioms as possible, and show that other properties are implied by this small list of axioms. Note, for example, that we have not yet introduced the operation of subtraction. (We will see this in ...
... statement is true, we may then use it in following statements. In general, we would like to assume as few axioms as possible, and show that other properties are implied by this small list of axioms. Note, for example, that we have not yet introduced the operation of subtraction. (We will see this in ...
Lecture Notes for MA 132 Foundations
... there is, to prove a new mathematical theorem, and that is what research mathematicians spend a lot of their time doing. It is also thought to be rather pedantic - the mathematician is the person who, where everyone else can see a brown cow, says “I see a cow, at least one side of which is brown”. T ...
... there is, to prove a new mathematical theorem, and that is what research mathematicians spend a lot of their time doing. It is also thought to be rather pedantic - the mathematician is the person who, where everyone else can see a brown cow, says “I see a cow, at least one side of which is brown”. T ...
Lecture notes, sections 2.5 to 2.7
... Let us now prove a few simple theorems, to show how congruences can be used. The important thing in these examples is not the statement of the theorem, but the method of proof, using congruences. Theorem 16. Every odd square gives the remainder 1 when divided by 4. Proof. (Recall that a square is an ...
... Let us now prove a few simple theorems, to show how congruences can be used. The important thing in these examples is not the statement of the theorem, but the method of proof, using congruences. Theorem 16. Every odd square gives the remainder 1 when divided by 4. Proof. (Recall that a square is an ...
4.2 Recursion, Recurrences and Induction
... element n is the same as the number of subsets without element n. The number of subsets without element n is just the number of subsets of an n − 1-element set. Thus the number of subsets of {1, 2, . . . , n} is twice the number of subsets of {1, 2, . . . , n − 1}. This will give us the inductive st ...
... element n is the same as the number of subsets without element n. The number of subsets without element n is just the number of subsets of an n − 1-element set. Thus the number of subsets of {1, 2, . . . , n} is twice the number of subsets of {1, 2, . . . , n − 1}. This will give us the inductive st ...
Short proofs of some extremal results
... deals with the problem of maximizing or minimizing the cardinality of a collection of finite objects satisfying a certain property. The problems we consider come mainly from the areas of extremal graph theory, Ramsey theory and additive combinatorics. In each case, we give a complete or partial solu ...
... deals with the problem of maximizing or minimizing the cardinality of a collection of finite objects satisfying a certain property. The problems we consider come mainly from the areas of extremal graph theory, Ramsey theory and additive combinatorics. In each case, we give a complete or partial solu ...
ICS 251 – Foundation of Computer Science – Fall 2002
... 34. Prove by induction that 1* 21 + 2 * 22 + ... + n*2n = (n-1) 2n+1 + 2. Proof: Basis Step: For n=1, LHS = 1* 21 = 2. RHS = (1-1) 21+1 + 2 = 2 = LHS. Induction Step: Assume P(n) and show P(n+1). Thus we assume that 1* 21 + 2 * 22 + ... + n*2n = (n-1) 2n+1 + 2, and show that 1* 21 + 2 * 22 + ... + n ...
... 34. Prove by induction that 1* 21 + 2 * 22 + ... + n*2n = (n-1) 2n+1 + 2. Proof: Basis Step: For n=1, LHS = 1* 21 = 2. RHS = (1-1) 21+1 + 2 = 2 = LHS. Induction Step: Assume P(n) and show P(n+1). Thus we assume that 1* 21 + 2 * 22 + ... + n*2n = (n-1) 2n+1 + 2, and show that 1* 21 + 2 * 22 + ... + n ...
euler and the partial sums of the prime
... Theorem 1 corresponds to the case F (s) = log ⇣(s). More sophisticated Tauberian theorems imply finer results about the distribution of primes. In fact, Tauberian theory furnishes what is arguably the simplest known approach to the prime number theorem. See, for example, the remarkably pithy exposit ...
... Theorem 1 corresponds to the case F (s) = log ⇣(s). More sophisticated Tauberian theorems imply finer results about the distribution of primes. In fact, Tauberian theory furnishes what is arguably the simplest known approach to the prime number theorem. See, for example, the remarkably pithy exposit ...
Partition of a Set which Contains an Infinite Arithmetic (Respectively
... (respectively geometric) progression into two subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression. Introduction. First, in this article we build sets which have the following property: for any par ...
... (respectively geometric) progression into two subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression. Introduction. First, in this article we build sets which have the following property: for any par ...
Lecture Notes - Department of Mathematics
... calculus class are tame. Let Ω be the set of all tame sets. (Ω being the last letter of the Greek alphabet is often used to denote “large” sets.) Is Ω tame? If it is, then it belongs to Ω and hence, Ω is wild. On the other hand, if Ω is wild then Ω is its element. Since by definition, all elements o ...
... calculus class are tame. Let Ω be the set of all tame sets. (Ω being the last letter of the Greek alphabet is often used to denote “large” sets.) Is Ω tame? If it is, then it belongs to Ω and hence, Ω is wild. On the other hand, if Ω is wild then Ω is its element. Since by definition, all elements o ...
![The full Müntz Theorem in C[0,1]](http://s1.studyres.com/store/data/019844035_1-f7b6943c075c22a54f0d5d67f46bc0e0-300x300.png)
The full Müntz Theorem in C[0,1]
... 1916, and anticipated by Bernstein [3] was only for sequences of exponents tending to infinity. Later works, see, for example, [22] and [19], include the above result, as well as a treatment of the case when {λi }∞ i=1 is a sequence of distinct, positive real numbers tending to 0. The novelty in thi ...
... 1916, and anticipated by Bernstein [3] was only for sequences of exponents tending to infinity. Later works, see, for example, [22] and [19], include the above result, as well as a treatment of the case when {λi }∞ i=1 is a sequence of distinct, positive real numbers tending to 0. The novelty in thi ...
Mathematical Knowledge for Teaching at the Secondary Level
... What is the sum of the kth powers of the first n natural numbers? ...
... What is the sum of the kth powers of the first n natural numbers? ...
