Chapter 1-sec1.1
... Check exhaustively and find that 211 is prime. QED. (quod erat demonstrandum) ...
... Check exhaustively and find that 211 is prime. QED. (quod erat demonstrandum) ...
Math 248, Methods of Proof, Winter 2015
... 3. Prove (by contradiction) that there does not exists a smallest positive real number (that is there does not exists an r ∈ R such that r > 0 and, if s ∈ R and s > 0 then r ≤ s). Sometimes we will want to prove that a statement of the form (∀x)(P (x)) is false. If we do this by giving a constructiv ...
... 3. Prove (by contradiction) that there does not exists a smallest positive real number (that is there does not exists an r ∈ R such that r > 0 and, if s ∈ R and s > 0 then r ≤ s). Sometimes we will want to prove that a statement of the form (∀x)(P (x)) is false. If we do this by giving a constructiv ...
Worksheet I: What is a proof (And what is not a proof)
... be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason, and the same argument shows that he cannot be hanged on any ...
... be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason, and the same argument shows that he cannot be hanged on any ...
PPT - School of Computer Science
... dark and they only have one flashlight, so people must cross either alone or in pairs (bringing the flashlight). Their walking speeds allow them to cross in 1, 2, 5, and 10 minutes, respectively. Is it possible for them to all cross in 17 minutes? ...
... dark and they only have one flashlight, so people must cross either alone or in pairs (bringing the flashlight). Their walking speeds allow them to cross in 1, 2, 5, and 10 minutes, respectively. Is it possible for them to all cross in 17 minutes? ...
Introduction to Algebraic Proof
... To prove that the product of three consecutive positive integers is a multiple of 6 we must choose the general case of three consecutive positive integers: k, k+1 and k+2 and use the properties of division of integers. First we check for the presence of even integers. The integer k can either be eve ...
... To prove that the product of three consecutive positive integers is a multiple of 6 we must choose the general case of three consecutive positive integers: k, k+1 and k+2 and use the properties of division of integers. First we check for the presence of even integers. The integer k can either be eve ...
Informal proofs
... Proving theorems in practice: • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often hel ...
... Proving theorems in practice: • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often hel ...
methods of proof
... Methods of Proving Theorems Proving mathematical theorems can be difficult. To construct proofs we need different proof methods. One we have chosen a proof method, we use axioms, definitions of terms, previously proved results, and rules of inference to complete the proof. To prove a theorem of the ...
... Methods of Proving Theorems Proving mathematical theorems can be difficult. To construct proofs we need different proof methods. One we have chosen a proof method, we use axioms, definitions of terms, previously proved results, and rules of inference to complete the proof. To prove a theorem of the ...
Practice Midterm 1
... (b) Why is the remaining statement NOT logically equivalent to the two logically equivalent statements in Part 4(a) ? (You must provide a concrete example where this statement is not logically equivalent to the two other statements.) 4. Give a concrete function f : Z → N that is injective but NOT su ...
... (b) Why is the remaining statement NOT logically equivalent to the two logically equivalent statements in Part 4(a) ? (You must provide a concrete example where this statement is not logically equivalent to the two other statements.) 4. Give a concrete function f : Z → N that is injective but NOT su ...
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.