Lecture 3
... • A term can be a constant, a variable or a function name applied to zero or more arguments e.g., add(X,Y). More complex terms can be built from a vocabulary of function symbols and variable symbols. Terms can be considered as simple strings. • Term rewriting is a computational method that is based ...
... • A term can be a constant, a variable or a function name applied to zero or more arguments e.g., add(X,Y). More complex terms can be built from a vocabulary of function symbols and variable symbols. Terms can be considered as simple strings. • Term rewriting is a computational method that is based ...
Solutions to selected homework problems
... which is divisible by 5 since n5 − n is divisible by 5 (by induction hypothesis). Problem: Show that every nonzero integer can be uniquely represented as: ek 3k + ek−1 3k−1 + · · · + e1 3 + e0 , where ej = −1, 0, 1 and ek 6= 0. Solution: To prove that any number can be represented this way just mimi ...
... which is divisible by 5 since n5 − n is divisible by 5 (by induction hypothesis). Problem: Show that every nonzero integer can be uniquely represented as: ek 3k + ek−1 3k−1 + · · · + e1 3 + e0 , where ej = −1, 0, 1 and ek 6= 0. Solution: To prove that any number can be represented this way just mimi ...
The strong law of large numbers - University of California, Berkeley
... be given in section 3, but they are all unsatisfactory. Presumably all (or shall we say most) mathematicians will agree on a satisfactory set of such conditions if and when they are exhibited, but before they are it does not seem easy to lay down criteria of satisfactoriness. On the other hand it is ...
... be given in section 3, but they are all unsatisfactory. Presumably all (or shall we say most) mathematicians will agree on a satisfactory set of such conditions if and when they are exhibited, but before they are it does not seem easy to lay down criteria of satisfactoriness. On the other hand it is ...
For a pdf file
... Clearly, n > p. Since p is the largest prime number, n cannot be a prime number. In other words, n is composite. Let q be any prime number. Because of the way n is constructed, when n is divided by q the remainder is 1. That is, n is not a multiple of q. This contradicts the Fundamental Theorem of A ...
... Clearly, n > p. Since p is the largest prime number, n cannot be a prime number. In other words, n is composite. Let q be any prime number. Because of the way n is constructed, when n is divided by q the remainder is 1. That is, n is not a multiple of q. This contradicts the Fundamental Theorem of A ...
Solution 4 - WUSTL Math
... Proof. We will show this by contradiction. So, assume the above statement is false. Then writing the negation, we see that we have a real number x, an irrational number y with both x + y and −x + y rational. Since the set of rational numbers is closed under addition, we see by adding, (x + y) + (−x ...
... Proof. We will show this by contradiction. So, assume the above statement is false. Then writing the negation, we see that we have a real number x, an irrational number y with both x + y and −x + y rational. Since the set of rational numbers is closed under addition, we see by adding, (x + y) + (−x ...
Friendly Logics, Fall 2015, Homework 1
... (a) Prove that the statement of the Soundness Theorem (3.1) is equivalent to the following statement: for any ⌃ if ⌃ is satisfiable then it is consistent. (b) Prove that the statement of the Compleness Theorem (3.3) is equivalent to the following statement: for any ⌃ if ⌃ is consistent then it is sa ...
... (a) Prove that the statement of the Soundness Theorem (3.1) is equivalent to the following statement: for any ⌃ if ⌃ is satisfiable then it is consistent. (b) Prove that the statement of the Compleness Theorem (3.3) is equivalent to the following statement: for any ⌃ if ⌃ is consistent then it is sa ...
Proof by contrapositive, contradiction
... be written as such a fraction, it can be written as a fraction in lowest terms, i.e. where a and b have no common factors. If a and b have common factors, it’s easy to remove them. Also, we proved last class that, for any integer k, if k is odd then k 2 is odd. So the contrapositive of this statemen ...
... be written as such a fraction, it can be written as a fraction in lowest terms, i.e. where a and b have no common factors. If a and b have common factors, it’s easy to remove them. Also, we proved last class that, for any integer k, if k is odd then k 2 is odd. So the contrapositive of this statemen ...
PROOFS Math 174 May 2017 I. Introduction. In the natural sciences
... I. Introduction. In the natural sciences, we cannot prove anything to be true. We can only support hypotheses with data and make decisions based on the most strongly-supported hypotheses, but the fact remains that fringe philosophical problems still apply (can we trust our senses? Are we living in a ...
... I. Introduction. In the natural sciences, we cannot prove anything to be true. We can only support hypotheses with data and make decisions based on the most strongly-supported hypotheses, but the fact remains that fringe philosophical problems still apply (can we trust our senses? Are we living in a ...
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.