7 : Induction
... theorems about the natural numbers. • The method is important in computing applications; it is closely related to recursion, and it is a useful tool if you are trying to establish that an algorithm is correct. ...
... theorems about the natural numbers. • The method is important in computing applications; it is closely related to recursion, and it is a useful tool if you are trying to establish that an algorithm is correct. ...
preprint - Open Science Framework
... CS1: If n is in the past, the subject inspects its perfect memory; if n is in the future, it can postpone its decision for the finite number of stages required and then check again. CS2: This just makes the presence of perfect memory explicit. When Brouwer made his first remark about the ideal mathe ...
... CS1: If n is in the past, the subject inspects its perfect memory; if n is in the future, it can postpone its decision for the finite number of stages required and then check again. CS2: This just makes the presence of perfect memory explicit. When Brouwer made his first remark about the ideal mathe ...
Chapter 1: The Foundations: Logic and Proofs Section 1.7
... a) a proof by contraposition. b) a proof by contradiction. Solution: 1. Proof by contraposition: We need to prove that if n is odd, then 3n + 2 is odd. Assume that n is odd. Then we can write n = 2k + 1 for some integer k. Then 3n + 2 = 3(2k + 1) + 2 = 6k + 5 = 2(3k + 2) + 1. – odd number Thus 3n + ...
... a) a proof by contraposition. b) a proof by contradiction. Solution: 1. Proof by contraposition: We need to prove that if n is odd, then 3n + 2 is odd. Assume that n is odd. Then we can write n = 2k + 1 for some integer k. Then 3n + 2 = 3(2k + 1) + 2 = 6k + 5 = 2(3k + 2) + 1. – odd number Thus 3n + ...
1 Cardinality and the Pigeonhole Principle
... Again, this is secretly a common sense definition, or at least one you’ve known since you were very young. When we want to know how many apples are in a bag and don’t have your standardized bags of oranges2 to compare them to we simply point at the apples one by one while uttering the words “one, tw ...
... Again, this is secretly a common sense definition, or at least one you’ve known since you were very young. When we want to know how many apples are in a bag and don’t have your standardized bags of oranges2 to compare them to we simply point at the apples one by one while uttering the words “one, tw ...
lecture1.5
... Disproving something: counterexamples If we are asked to show that a proposition is False, then we just need to provide one counter-example for which the proposition is False In other words, to show that x P(x) is False, we can just show x P(x) = x P(x) to be True Example: “Every positive inte ...
... Disproving something: counterexamples If we are asked to show that a proposition is False, then we just need to provide one counter-example for which the proposition is False In other words, to show that x P(x) is False, we can just show x P(x) = x P(x) to be True Example: “Every positive inte ...
6/FORMAL PROOF OF VALIDITY
... A>-C by Transposition (line 5) and by Double Negation (line 6) so that we can proceed with A>C (line 4) and C>-A (line 6)by Hypo thetical Syllogism. These rules help facilitate the construction of proofs. The second example is more complicated, and as can be observed, no inference rules would allow ...
... A>-C by Transposition (line 5) and by Double Negation (line 6) so that we can proceed with A>C (line 4) and C>-A (line 6)by Hypo thetical Syllogism. These rules help facilitate the construction of proofs. The second example is more complicated, and as can be observed, no inference rules would allow ...
9.6 Mathematical Induction
... The Tower of Hanoi Problem You might be familiar with a game that is played with a stack of round washers of different diameters and a stand with three vertical pegs (Figure 9.11). The game is not difficult to win once you get the hang of it, but it takes a while to move all the washers even when yo ...
... The Tower of Hanoi Problem You might be familiar with a game that is played with a stack of round washers of different diameters and a stand with three vertical pegs (Figure 9.11). The game is not difficult to win once you get the hang of it, but it takes a while to move all the washers even when yo ...
Trees
... Proof by Induction Prove a statement for a set of objects Base: prove it for a “small” object Induction: prove it for “bigger” objects assuming it holds for “smaller” ones Natural numbers Inductively defined objects ...
... Proof by Induction Prove a statement for a set of objects Base: prove it for a “small” object Induction: prove it for “bigger” objects assuming it holds for “smaller” ones Natural numbers Inductively defined objects ...
Math 248, Methods of Proof, Winter 2015
... 6. Prove that for all x ∈ R, |x| ≥ 0. Prove this as though you know how to do usual arithmetic on R (and in particular that −a > −b if and only if a < b), but are just learning about |x| for the first time. Of course, when encountered in the wild, a proof might require more than just two cases. 7. P ...
... 6. Prove that for all x ∈ R, |x| ≥ 0. Prove this as though you know how to do usual arithmetic on R (and in particular that −a > −b if and only if a < b), but are just learning about |x| for the first time. Of course, when encountered in the wild, a proof might require more than just two cases. 7. P ...
Section 1.5 Proofs in Predicate Logic
... theorem false; i.e. there does not exist a real number x such that for all real numbers y the equation x + y = 1 holds. As a counterexample select x = 2 , then clearly 2 + y = 1 is not true for all y ( y = 3 fails). Proofs by Contradiction for Universal and Existential Quantifiers Proof of ( ∀x ) P ...
... theorem false; i.e. there does not exist a real number x such that for all real numbers y the equation x + y = 1 holds. As a counterexample select x = 2 , then clearly 2 + y = 1 is not true for all y ( y = 3 fails). Proofs by Contradiction for Universal and Existential Quantifiers Proof of ( ∀x ) P ...
Axiomatic Systems
... Axiom 1: Each silly is a set of exactly three dillies. Axiom 2: There are exactly four dillies. Axiom 3: Each dilly is contained in a silly. Axiom 4: No dilly is contained in more than one silly. Notice that in the second example, the axioms defined a new term (“identity”). This isn’t an undefined t ...
... Axiom 1: Each silly is a set of exactly three dillies. Axiom 2: There are exactly four dillies. Axiom 3: Each dilly is contained in a silly. Axiom 4: No dilly is contained in more than one silly. Notice that in the second example, the axioms defined a new term (“identity”). This isn’t an undefined t ...
Week 1: Logic Lecture 1, 8/21 (Sections 1.1 and 1.3)
... when you plug in a value for x, and then has a truth value. • Note: I’ll say much more about functions in the next two weeks. • Propositional functions can have many input variables, such as the statement “x = y + 3”, which we could denote Q(x, y). (Think of values for x and y making the statement t ...
... when you plug in a value for x, and then has a truth value. • Note: I’ll say much more about functions in the next two weeks. • Propositional functions can have many input variables, such as the statement “x = y + 3”, which we could denote Q(x, y). (Think of values for x and y making the statement t ...
Solution - Stony Brook Mathematics
... same sex as the people in the middle, who in turn are of the same sex as ...
... same sex as the people in the middle, who in turn are of the same sex as ...
homework
... (a) Is there a 1-1 correspondence between the nodes of this tree and the natural numbers? (b) Is there a 1-1 correspondence between the branches of this tree and the natural numbers? (Don’t forget to justify your answers!) Note: A branch is an infinite sequence of nodes which starts at the top of th ...
... (a) Is there a 1-1 correspondence between the nodes of this tree and the natural numbers? (b) Is there a 1-1 correspondence between the branches of this tree and the natural numbers? (Don’t forget to justify your answers!) Note: A branch is an infinite sequence of nodes which starts at the top of th ...
Full text
... The first section of "Pythagorean Triangles" is primarily a portion of the history of Pythagorean triangles and related problems. However, some new results and some new proofs of old results are presented in this section. For example, Fermat's Theorem is used to prove: Levy's Theorem, If (x,y,z) is ...
... The first section of "Pythagorean Triangles" is primarily a portion of the history of Pythagorean triangles and related problems. However, some new results and some new proofs of old results are presented in this section. For example, Fermat's Theorem is used to prove: Levy's Theorem, If (x,y,z) is ...
CHAP03 Induction and Finite Series
... The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … Now notice this pattern. The number 02 + 0 + 41 = 41, which is a prime number. The number 12 + 1 + 41 = 43, which is a prime number. The number 22 + 2 + 41 = 47, which is a prime number. The number 32 + 3 + 41 = 53, which is a pri ...
... The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … Now notice this pattern. The number 02 + 0 + 41 = 41, which is a prime number. The number 12 + 1 + 41 = 43, which is a prime number. The number 22 + 2 + 41 = 47, which is a prime number. The number 32 + 3 + 41 = 53, which is a pri ...
Document
... A formal description of mathematical induction can be illustrated by reference to the sequential effect of falling dominoes ...
... A formal description of mathematical induction can be illustrated by reference to the sequential effect of falling dominoes ...
