Mathematics in Context Sample Review Questions
... From this triangle construct two squares with sides of length a + b, also shown above. The two squares have the same lengths for their sides, so their areas must be equal. a. Calculate the area of the left-hand square by adding up the areas of the triangles and squares that compose it. (4) ...
... From this triangle construct two squares with sides of length a + b, also shown above. The two squares have the same lengths for their sides, so their areas must be equal. a. Calculate the area of the left-hand square by adding up the areas of the triangles and squares that compose it. (4) ...
The Non-Euclidean Revolution Material Axiomatic Systems and the
... The Non-Euclidean Revolution Material Axiomatic Systems and the Turtle Club Example Recall that a material axiomatic system consists of four parts: the primitive (or undefined) terms, the defined terms, the axioms (or assumptions) used as the starting point for deduction, and the theorems (or statem ...
... The Non-Euclidean Revolution Material Axiomatic Systems and the Turtle Club Example Recall that a material axiomatic system consists of four parts: the primitive (or undefined) terms, the defined terms, the axioms (or assumptions) used as the starting point for deduction, and the theorems (or statem ...
Discrete Structures & Algorithms Propositional Logic
... i.e., that n is even. Then n=2k for some integer k. Then 3n+2 = 3(2k)+2 = 6k+2 = 2(3k+1). Thus 3n+2 is even, because it equals 2j for integer j = 3k+1. So 3n+2 is not odd. We have shown that ¬(n is odd)→¬(3n+2 is odd), thus its contra-positive (3n+2 is odd) → (n is odd) is also true. □ ...
... i.e., that n is even. Then n=2k for some integer k. Then 3n+2 = 3(2k)+2 = 6k+2 = 2(3k+1). Thus 3n+2 is even, because it equals 2j for integer j = 3k+1. So 3n+2 is not odd. We have shown that ¬(n is odd)→¬(3n+2 is odd), thus its contra-positive (3n+2 is odd) → (n is odd) is also true. □ ...
Full text
... not mention skew circulants. His proof depends on Gauss's theory of "biquadrati c residues" and on work by Lagrange (presumably Lagrange's Oeuvres III, pp. 693-795). Jacobifs proof is too succinct for me to understand, and I think he may have been slightly careless. For example, he says (in free tra ...
... not mention skew circulants. His proof depends on Gauss's theory of "biquadrati c residues" and on work by Lagrange (presumably Lagrange's Oeuvres III, pp. 693-795). Jacobifs proof is too succinct for me to understand, and I think he may have been slightly careless. For example, he says (in free tra ...
25.4 Sum-product sets
... 0 so that max{|A + A|, |A · A|} ≥ |A|1+δ for all sufficiently large sets A. This parameter δ has been steadily improved by a number of authors. One highlight in this sequence is a proof by Elekes (1997) that δ may be taken arbitrarily close to 1/4. His argument utilizes a clever application of the Sze ...
... 0 so that max{|A + A|, |A · A|} ≥ |A|1+δ for all sufficiently large sets A. This parameter δ has been steadily improved by a number of authors. One highlight in this sequence is a proof by Elekes (1997) that δ may be taken arbitrarily close to 1/4. His argument utilizes a clever application of the Sze ...
Logic and Proof - Collaboratory for Advanced Computing and
... have no common factors)”. • It follows that 2 = a2/b2, hence 2b2 = a2. This means a2 is even, which implies a is even. Furthermore, since a is even, a = 2c for some integer c. Thus 2b2 = 4c2, so b2 = 2c2. This means b2 is even, which implies b is even. • It has been shown that ¬p → “√2 = a/b, wh ...
... have no common factors)”. • It follows that 2 = a2/b2, hence 2b2 = a2. This means a2 is even, which implies a is even. Furthermore, since a is even, a = 2c for some integer c. Thus 2b2 = 4c2, so b2 = 2c2. This means b2 is even, which implies b is even. • It has been shown that ¬p → “√2 = a/b, wh ...
Introduction - Computer Science
... algorithms, data structures, database, parallel computing, distributed systems, cryptography, computer networks… ...
... algorithms, data structures, database, parallel computing, distributed systems, cryptography, computer networks… ...
Irrationality of Square Roots - Mathematical Association of America
... where c0 , . . . , cn−1 are integers, would have to be ≥ 1/q n−1 . We can construct arbitrarily small positive expressions of the form (3) by expanding (α − α)k in powers of α and eliminating terms with exponents ≥ n by repeated use of α n = −an−1 α n−1 − · · · − a0 . We conclude that α is irratio ...
... where c0 , . . . , cn−1 are integers, would have to be ≥ 1/q n−1 . We can construct arbitrarily small positive expressions of the form (3) by expanding (α − α)k in powers of α and eliminating terms with exponents ≥ n by repeated use of α n = −an−1 α n−1 − · · · − a0 . We conclude that α is irratio ...
Comparing Contrapositive and Contradiction Proofs
... Only works for a finite number of cases. The standard approach to try. Contrapositive Assume Q', deduce P' Use if Q' as a (Indirect) Proof hypothesis seems to give more information to work with. Contradiction Assume P Λ Q', deduce Try this approach when a contradiction Q says something is not true. ...
... Only works for a finite number of cases. The standard approach to try. Contrapositive Assume Q', deduce P' Use if Q' as a (Indirect) Proof hypothesis seems to give more information to work with. Contradiction Assume P Λ Q', deduce Try this approach when a contradiction Q says something is not true. ...
Contradiction: means to follow a path toward which a statement
... If and only if (↔) : In order to prove this you must prove p→q and q→p (p←q) are both true using the above methods. Uniqueness: Is generally performed by contradiction. I may involve solving for the unique value and then trying to produce similar values by the same method that leads to a contraction ...
... If and only if (↔) : In order to prove this you must prove p→q and q→p (p←q) are both true using the above methods. Uniqueness: Is generally performed by contradiction. I may involve solving for the unique value and then trying to produce similar values by the same method that leads to a contraction ...
SESSION 1: PROOF 1. What is a “proof”
... counterexample to the above statement, and we would conclude that the above statement is false; namely there exist integers n, x, y, z such that xn + y n = z n . Indeed, n = x = y = 1, and z = 2 is a counterexample to the above statement 1. 2.3. Proof by contradiction. Suppose that we would like to ...
... counterexample to the above statement, and we would conclude that the above statement is false; namely there exist integers n, x, y, z such that xn + y n = z n . Indeed, n = x = y = 1, and z = 2 is a counterexample to the above statement 1. 2.3. Proof by contradiction. Suppose that we would like to ...
m5zn_8a0e185bfba5c83
... (2) Non-constructive Existence Proof The proof is established by showing that an object a with P(a) is true must exist without explicitly demonstrating one. Proofs by contradiction are usually used in such cases. Example: Let x1,x2,..,xn be positive integers such that their average is m. prove t ...
... (2) Non-constructive Existence Proof The proof is established by showing that an object a with P(a) is true must exist without explicitly demonstrating one. Proofs by contradiction are usually used in such cases. Example: Let x1,x2,..,xn be positive integers such that their average is m. prove t ...
ppt
... Gödel’s Proof G: This statement of number theory does not have any proof in the system of PM. If G were provable, then PM would be inconsistent. If G is unprovable, then PM would be incomplete. PM cannot be complete and consistent! ...
... Gödel’s Proof G: This statement of number theory does not have any proof in the system of PM. If G were provable, then PM would be inconsistent. If G is unprovable, then PM would be incomplete. PM cannot be complete and consistent! ...
Proofs and Proof Methods
... The Forward-Backward Method and Indirect Proof • Once we have established our premises and our conclusion, we need to construct a chain of equivalences and implications that lead from the premises to the conclusion. To search for this, we might try to find the first step in the chain, by finding so ...
... The Forward-Backward Method and Indirect Proof • Once we have established our premises and our conclusion, we need to construct a chain of equivalences and implications that lead from the premises to the conclusion. To search for this, we might try to find the first step in the chain, by finding so ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.