Fraïssé`s conjecture in Pi^1_1-comprehension
... FRA: The class of countable linear orderings is well-quasi-ordered under embeddability. A well-quasi-ordering is a quasi-ordering without infinite descending sequences or infinite antichains. The study of well-quasi-orderings has been of interest to reverse mathematicians for a long time because it ...
... FRA: The class of countable linear orderings is well-quasi-ordered under embeddability. A well-quasi-ordering is a quasi-ordering without infinite descending sequences or infinite antichains. The study of well-quasi-orderings has been of interest to reverse mathematicians for a long time because it ...
CS 381 Midterm Review
... 14. Compute the ceiling and floor of any real number. 15. Write pseudocode to solve a list-based problem. 16. Given pseudocode for an algorithm, write a function that calculates the number of times a particular line of the algorithm is executed. 17. Given a function, determine and prove big-O, big-O ...
... 14. Compute the ceiling and floor of any real number. 15. Write pseudocode to solve a list-based problem. 16. Given pseudocode for an algorithm, write a function that calculates the number of times a particular line of the algorithm is executed. 17. Given a function, determine and prove big-O, big-O ...
Extra Examples — Page references correspond to locations of Extra
... to show that neither root can be written as a quotient of integers. It might be easier to try a proof by contradiction because we can make the assumption that the roots are rational and hence can be written as fractions. That is, there is a very specific form in which the roots can be written. Suppos ...
... to show that neither root can be written as a quotient of integers. It might be easier to try a proof by contradiction because we can make the assumption that the roots are rational and hence can be written as fractions. That is, there is a very specific form in which the roots can be written. Suppos ...
The Future of Post-Human Mathematical Logic
... (and Other Mental States) ............................................................67 Table 1.11. The Theoretical Levels of Consciousness (and Other Mental States) ............................................................68 Table 1.12. The Thematic Issues of Consciousness (and Other Mental Sta ...
... (and Other Mental States) ............................................................67 Table 1.11. The Theoretical Levels of Consciousness (and Other Mental States) ............................................................68 Table 1.12. The Thematic Issues of Consciousness (and Other Mental Sta ...
HW 7. - U.I.U.C. Math
... Puzzle of the Week: Here is another game-type puzzle involving a “magic number” that you can try on a friend. It is not particularly difficult, it is not hard to guess a formula for magic number, and proving that this number has the desired “magic” properties is not that hard either. #1 A magic matr ...
... Puzzle of the Week: Here is another game-type puzzle involving a “magic number” that you can try on a friend. It is not particularly difficult, it is not hard to guess a formula for magic number, and proving that this number has the desired “magic” properties is not that hard either. #1 A magic matr ...
Homomorphism Preservation Theorem
... Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction: ...
... Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction: ...
Midterm Exam 2 Solutions, Comments, and Feedback
... obtained statement P (k + 1). Hence, the proof of the induction step is complete. Conclusion: By the principle of induction, P (n) is true for all n ∈ N. 5. Let A, B, C be sets, f : A → B, and g : B → C be functions, and let h : A → C be defined by h(x) = g(f (x)) for x ∈ A. (a) Give a careful, step ...
... obtained statement P (k + 1). Hence, the proof of the induction step is complete. Conclusion: By the principle of induction, P (n) is true for all n ∈ N. 5. Let A, B, C be sets, f : A → B, and g : B → C be functions, and let h : A → C be defined by h(x) = g(f (x)) for x ∈ A. (a) Give a careful, step ...
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.