On mathematical induction
... Example 4.2. Let A be a set with n elements. Then the set P(A) of all subsets of A (the set P(A) is called the power set of A) contains 2n elements. Proof. We prove the theorem by induction over n. First we check the case in which A contains 0 elements. In this case A is the empty set ∅ and his only ...
... Example 4.2. Let A be a set with n elements. Then the set P(A) of all subsets of A (the set P(A) is called the power set of A) contains 2n elements. Proof. We prove the theorem by induction over n. First we check the case in which A contains 0 elements. In this case A is the empty set ∅ and his only ...
On the computational content of intuitionistic propositional proofs
... B is intuitionistically valid. A classical result of Harrop generalizes this result to sequents Γ→A ∨ B where Γ is a set of so called Harrop formulas. These are defined by: 1. every atomic formula is Harrop, ⊥ is Harrop; 2. if A and B are Harrop, then A ∧ B is Harrop; 3. if A is arbitrary and B is H ...
... B is intuitionistically valid. A classical result of Harrop generalizes this result to sequents Γ→A ∨ B where Γ is a set of so called Harrop formulas. These are defined by: 1. every atomic formula is Harrop, ⊥ is Harrop; 2. if A and B are Harrop, then A ∧ B is Harrop; 3. if A is arbitrary and B is H ...
ZENO`S PARADOX – THEOREM AND PROOF 1
... put, Zeno’s paradox claims that it is impossible to travel from point A to point B, since an infinite number of halves exist between the two points. For instance, a rock cannot fall from a height of 10 feet to the ground of zero feet, because the first half is five feet, the next half is 2.5 feet an ...
... put, Zeno’s paradox claims that it is impossible to travel from point A to point B, since an infinite number of halves exist between the two points. For instance, a rock cannot fall from a height of 10 feet to the ground of zero feet, because the first half is five feet, the next half is 2.5 feet an ...
(1) (a) Prove that if an integer n has the form 6q + 5 for some q ∈ Z
... (5) Consider the statement, the sum of any three consecutive positive perfect cubes is divisible by 9. (a) Sum the cubes of 4, 5 and 6 and verify that the resulting number is divisible by 9. (b) Prove the statement using mathematical induction. (c) Prove the statement by considering three cases, dep ...
... (5) Consider the statement, the sum of any three consecutive positive perfect cubes is divisible by 9. (a) Sum the cubes of 4, 5 and 6 and verify that the resulting number is divisible by 9. (b) Prove the statement using mathematical induction. (c) Prove the statement by considering three cases, dep ...
CSci 2011 Discrete Mathematics
... Axiom: A statement that is assumed to be true. Lemma: a less important theorem that is useful to prove a theorem. Corollary: a theorem that can be proven directly from a theorem that has been proved. Conjecture: a statement that is being proposed to be a true statement. ...
... Axiom: A statement that is assumed to be true. Lemma: a less important theorem that is useful to prove a theorem. Corollary: a theorem that can be proven directly from a theorem that has been proved. Conjecture: a statement that is being proposed to be a true statement. ...
Assignment MCS-013 Discrete Mathematics Q1: a) Make truth table
... (i) p(m) is true for some m belong N (ii) whenever p(m) ,p(m+1),………,p(k) are true, That p(k+1) is true, where k>=m we can conclude that p(n) is true for all natural number n>=m. In the induction step we are making more assumption. That p(n) is true for every n lying between m and k , not just that p ...
... (i) p(m) is true for some m belong N (ii) whenever p(m) ,p(m+1),………,p(k) are true, That p(k+1) is true, where k>=m we can conclude that p(n) is true for all natural number n>=m. In the induction step we are making more assumption. That p(n) is true for every n lying between m and k , not just that p ...
A 1 ∪A 2 ∪…∪A n |=|A 1 |+|A 2 |+…+|A n
... Proof: In constructing a r-permutation of S, we can choose the first item can be an object of any one of the k types. Similarly the second item to be an object of any one of the k types, and so on. Since all repetition numbers of S are infinite, the number of different choices for any item i ...
... Proof: In constructing a r-permutation of S, we can choose the first item can be an object of any one of the k types. Similarly the second item to be an object of any one of the k types, and so on. Since all repetition numbers of S are infinite, the number of different choices for any item i ...
1.4 Deductive Reasoning
... Inductive*reasoning*is*not*a*proof*of*anything*except*for*possibilities*that*you*tested.* There*could*always*be*a*counterexample*just*around*the*corner.* ...
... Inductive*reasoning*is*not*a*proof*of*anything*except*for*possibilities*that*you*tested.* There*could*always*be*a*counterexample*just*around*the*corner.* ...
Induction
... We prove by induction on the number of horses that all horses have the same color. The smallest group of horses is a group of one horse. Since we usually write our predicate P so that P (0) corresponds to the base case, we define P (n) as “All horses in a group of n + 1 horses have the same color.” ...
... We prove by induction on the number of horses that all horses have the same color. The smallest group of horses is a group of one horse. Since we usually write our predicate P so that P (0) corresponds to the base case, we define P (n) as “All horses in a group of n + 1 horses have the same color.” ...
If T is a consistent theory in the language of arithmetic, we say a set
... This is, in fact, just the formal calculation exhibited in section 6.1. Obviously this method is perfectly general, and whenever a + b = c we can prove a + b = c. Then also, again as in section 6.1, the recursion equations (Q5) and (Q6) for multiplication can be used to prove 2 · 3 = 6 and more gene ...
... This is, in fact, just the formal calculation exhibited in section 6.1. Obviously this method is perfectly general, and whenever a + b = c we can prove a + b = c. Then also, again as in section 6.1, the recursion equations (Q5) and (Q6) for multiplication can be used to prove 2 · 3 = 6 and more gene ...
KRIPKE-PLATEK SET THEORY AND THE ANTI
... 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are ...
... 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are ...
Chap 1/2 Review KEY
... 8. Ali created a math trick in which she always ended with 4. When Ali tried to prove her trick, however, it did not work. Ali’s Proof: n I used n to represent any number. 2n Multiply by 2. 2n + 8 Add 8. 2n + 4 Divide by 2. n+4 Subtract your starting number. Identify the error in Ali’s proof, and ex ...
... 8. Ali created a math trick in which she always ended with 4. When Ali tried to prove her trick, however, it did not work. Ali’s Proof: n I used n to represent any number. 2n Multiply by 2. 2n + 8 Add 8. 2n + 4 Divide by 2. n+4 Subtract your starting number. Identify the error in Ali’s proof, and ex ...
The Impossibility of Trisecting an Angle with Straightedge and
... and compass. That is, given an angle θ , give a procedure using only straightedge and compass that will construct the angle θ 3 in a finite number of steps. This problem, which dates to around 400 B.C., fascinated mathematicians and amateurs over the centuries and many “solutions” have been proposed ...
... and compass. That is, given an angle θ , give a procedure using only straightedge and compass that will construct the angle θ 3 in a finite number of steps. This problem, which dates to around 400 B.C., fascinated mathematicians and amateurs over the centuries and many “solutions” have been proposed ...
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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.