1.4 The set of Real Numbers: Quick Definition and
... belong to a real analysis class. Students should have studied (or will study) in a modern algebra class these properties and how to prove them. They are given here because they are essential to the everyday manipulations we perform when we work with real numbers. However, we will not spend time on t ...
... belong to a real analysis class. Students should have studied (or will study) in a modern algebra class these properties and how to prove them. They are given here because they are essential to the everyday manipulations we perform when we work with real numbers. However, we will not spend time on t ...
Ramsey`s Theorem and Compactness
... If f is an n-coloring of X, a subset Y ⊆ X is homogeneous or monochromatic for f if there is some i ∈ C such that, for every s ∈ Y [n] , we have f (s) = i. We call Y monochromatic because all n-element subsets of Y have the same color i. In the terms of this definition, the fact we proved in Section ...
... If f is an n-coloring of X, a subset Y ⊆ X is homogeneous or monochromatic for f if there is some i ∈ C such that, for every s ∈ Y [n] , we have f (s) = i. We call Y monochromatic because all n-element subsets of Y have the same color i. In the terms of this definition, the fact we proved in Section ...
Discrete Mathematics I Lectures Chapter 4
... be as messy as youtheorem, etc. By definition, hypothesis, by previous like! Use good connecting words to guide logic •You don’t have to use complete Then, thus, so, hence, It follows that, therefore, etc. sentences, give reasons, or even be clear DISPLAY equations and inequalities about assumpt ...
... be as messy as youtheorem, etc. By definition, hypothesis, by previous like! Use good connecting words to guide logic •You don’t have to use complete Then, thus, so, hence, It follows that, therefore, etc. sentences, give reasons, or even be clear DISPLAY equations and inequalities about assumpt ...
Direct proof
... Prove: If n is an integer and 3n + 2 is odd, then n is odd. Proof: Assume that 3n + 2 is odd and n is even (i.e., n = 2k) 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1) The above statement tells us that 3n + 2 is even, which is a contradiction of our assumption that 3n + 2 is odd. Therefore, we hav ...
... Prove: If n is an integer and 3n + 2 is odd, then n is odd. Proof: Assume that 3n + 2 is odd and n is even (i.e., n = 2k) 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1) The above statement tells us that 3n + 2 is even, which is a contradiction of our assumption that 3n + 2 is odd. Therefore, we hav ...
Number theory and proof techniques
... n and asks to find factors of it Integers r and s such that rs=n How would you do it? Can try all integers from 1 to n Can try all integers from 1 to sqrt(n) Can try all primes from 1 to sqrt(n) ...
... n and asks to find factors of it Integers r and s such that rs=n How would you do it? Can try all integers from 1 to n Can try all integers from 1 to sqrt(n) Can try all primes from 1 to sqrt(n) ...
Methods of Proof
... n is even. Then we can express n as 2k, where k is an integer. Therefore 3n+2 is then 6k+2, i.e. 2(3k+1), and this is an even number. This contradicts our assumptions, consequently n must be odd. Therefore when 3n+2 is odd, n is odd. QED ...
... n is even. Then we can express n as 2k, where k is an integer. Therefore 3n+2 is then 6k+2, i.e. 2(3k+1), and this is an even number. This contradicts our assumptions, consequently n must be odd. Therefore when 3n+2 is odd, n is odd. QED ...
Simple Continued Fractions for Some Irrational Numbers
... = B(u, v + 1), as was to be shown. (CF3) ensures the uniqueness of the result. Note that the continued fraction for B(u, v + 1) given in (10) has a total of 2n + 1 partial denominators while the continued fraction for B(u, v) has n + 1 partial denominators. We may now justify our assumption that n i ...
... = B(u, v + 1), as was to be shown. (CF3) ensures the uniqueness of the result. Note that the continued fraction for B(u, v + 1) given in (10) has a total of 2n + 1 partial denominators while the continued fraction for B(u, v) has n + 1 partial denominators. We may now justify our assumption that n i ...
ch02s1
... Used when exhaustive proof doesn’t work. Using the rules of propositional and predicate logic, prove P Q. Hence, assume the hypothesis P and prove Q. Hence, a formal proof would consist of a proof sequence to go from P to Q. Consider the conjecture x is an even integer Λ y is an even integer the ...
... Used when exhaustive proof doesn’t work. Using the rules of propositional and predicate logic, prove P Q. Hence, assume the hypothesis P and prove Q. Hence, a formal proof would consist of a proof sequence to go from P to Q. Consider the conjecture x is an even integer Λ y is an even integer the ...
Section 1
... Used when exhaustive proof doesn’t work. Using the rules of propositional and predicate logic, prove P Q. Hence, assume the hypothesis P and prove Q. Hence, a formal proof would consist of a proof sequence to go from P to Q. Consider the conjecture x is an even integer Λ y is an even integer the ...
... Used when exhaustive proof doesn’t work. Using the rules of propositional and predicate logic, prove P Q. Hence, assume the hypothesis P and prove Q. Hence, a formal proof would consist of a proof sequence to go from P to Q. Consider the conjecture x is an even integer Λ y is an even integer the ...
(pdf)
... published posthumously in 1930, to have applications to mathematical logic. After Ramsey, many famous mathematicians worked on what we now call Ramsey theory, including Erdos. Results in Ramsey theory are tied together by what Landman and Robertson describe as, “The study of the preservation of prop ...
... published posthumously in 1930, to have applications to mathematical logic. After Ramsey, many famous mathematicians worked on what we now call Ramsey theory, including Erdos. Results in Ramsey theory are tied together by what Landman and Robertson describe as, “The study of the preservation of prop ...
MATHEMATICAL INDUCTION
... exposita. The exact procedure need not concern us here. We only mention that one of the axioms was so designed as to incorporate induction as a method of proof. In other words, the intuitive way to deal with induction below is actually a legitimate technique. In what follows, the theory is presented ...
... exposita. The exact procedure need not concern us here. We only mention that one of the axioms was so designed as to incorporate induction as a method of proof. In other words, the intuitive way to deal with induction below is actually a legitimate technique. In what follows, the theory is presented ...
2.5-updated - WordPress.com
... What is the 50th string in the sequence? (1) – string of length 0 (2) (3) – string of length 1 (4) (7) string of length 2 (8) (15) – string of length 3 (16) (31) string of length 4 (32) (63) string of length 5 Hence the 50th string is of length 5. Among these (32) to (47 ...
... What is the 50th string in the sequence? (1) – string of length 0 (2) (3) – string of length 1 (4) (7) string of length 2 (8) (15) – string of length 3 (16) (31) string of length 4 (32) (63) string of length 5 Hence the 50th string is of length 5. Among these (32) to (47 ...
I(k-1)
... integers (or a subset like integers larger than 3) that have appropriate self-referential structure— including both equalities and inequalities—using either weak or strong induction as needed. – Critique formal inductive proofs to determine whether they are valid and where the error(s) lie if they a ...
... integers (or a subset like integers larger than 3) that have appropriate self-referential structure— including both equalities and inequalities—using either weak or strong induction as needed. – Critique formal inductive proofs to determine whether they are valid and where the error(s) lie if they a ...
Proof methods and Strategy
... At last step, 1,2 or 3 stones are left on the pile. (How can player 1 make player 2 leave 1, 2 or 3 stones on the pile?) Player 1 leaves 4 stones on the pile. (How many stones should be left on the pile for player 1?) 5, 6 or 7 stones are left on the pile for player 1. (How can player 1 make p ...
... At last step, 1,2 or 3 stones are left on the pile. (How can player 1 make player 2 leave 1, 2 or 3 stones on the pile?) Player 1 leaves 4 stones on the pile. (How many stones should be left on the pile for player 1?) 5, 6 or 7 stones are left on the pile for player 1. (How can player 1 make p ...
Arithmetic in Metamath, Case Study: Bertrand`s Postulate
... arithmetic on Q can be expressed in the language of Peano Arithmetic. Section 2 describes the metatheory of “decimal numbers” as they appear in set.mm, and the theorems which form the basic operations upon which the algorithm is built. Section 3 describes the Mathematica implementation of a limited- ...
... arithmetic on Q can be expressed in the language of Peano Arithmetic. Section 2 describes the metatheory of “decimal numbers” as they appear in set.mm, and the theorems which form the basic operations upon which the algorithm is built. Section 3 describes the Mathematica implementation of a limited- ...
4 The Natural Numbers
... The next topic we consider is the set-theoretic reconstruction of the theory of natural numbers. This is a key part of the general program to reduce mathematics to set theory. The basic strategy is to reduce classical arithmetic (thought of as the theory of the natural numbers) to set theory, and ha ...
... The next topic we consider is the set-theoretic reconstruction of the theory of natural numbers. This is a key part of the general program to reduce mathematics to set theory. The basic strategy is to reduce classical arithmetic (thought of as the theory of the natural numbers) to set theory, and ha ...
Writing Proofs
... is true.” We’ll discuss several of them in these pages. It may not be obvious at first which variety of proof to use, but a good rule of thumb is to try a direct proof first. A direct proof. Start by assuming that statement A is true. After all, if statement A is false then there’s nothing to worry ...
... is true.” We’ll discuss several of them in these pages. It may not be obvious at first which variety of proof to use, but a good rule of thumb is to try a direct proof first. A direct proof. Start by assuming that statement A is true. After all, if statement A is false then there’s nothing to worry ...
3.3 Proofs Involving Quantifiers 1. In exercise 6 of Section 2.2 you
... since the goal is to find x such that ∀y ∈ R(xy 2 = y − x), we cannot start the proof by assigning some value to x. 18. Consider the following incorrect theorem: Incorrect Theorem. Suppose F and G are families of sets. If ∪F and ∪G are disjoint, then so are F and G. (a) What’s wrong with the followi ...
... since the goal is to find x such that ∀y ∈ R(xy 2 = y − x), we cannot start the proof by assigning some value to x. 18. Consider the following incorrect theorem: Incorrect Theorem. Suppose F and G are families of sets. If ∪F and ∪G are disjoint, then so are F and G. (a) What’s wrong with the followi ...