Computability

... [Note: Sipser uses size.] Georg Cantor (1873) noticed that two finite sets are the same size if the elements in each can be paired. Definition: Two sets A & B have the same cardinality (size?) if there exists a function f: A → B, that is 1 to 1 and onto 1 to 1 means: if f(a) = f(b) then a=b. onto me ...

... [Note: Sipser uses size.] Georg Cantor (1873) noticed that two finite sets are the same size if the elements in each can be paired. Definition: Two sets A & B have the same cardinality (size?) if there exists a function f: A → B, that is 1 to 1 and onto 1 to 1 means: if f(a) = f(b) then a=b. onto me ...

COMP 9 / EN47 - Exploring Computer Science

... Some problems cannot be solved! Problem ‘A’: while x != 1 do x = x - 2 Clearly, this only terminates for even values of x Problem ‘B’: while x != 1 do if x.even? x = x / 2 else x = 3 * x + 1 end This terminates for some values of x, but nobody has been able to prove it terminates for all! An example ...

... Some problems cannot be solved! Problem ‘A’: while x != 1 do x = x - 2 Clearly, this only terminates for even values of x Problem ‘B’: while x != 1 do if x.even? x = x / 2 else x = 3 * x + 1 end This terminates for some values of x, but nobody has been able to prove it terminates for all! An example ...

slides - Center for Collective Dynamics of Complex Systems (CoCo)

... Universal Turing machines (UTM) • More important fact shown by Turing: There are TMs that can emulate behaviors of any other TMs if instructions are given (software) A specific TM can be computationally universal just by itself! ...

... Universal Turing machines (UTM) • More important fact shown by Turing: There are TMs that can emulate behaviors of any other TMs if instructions are given (software) A specific TM can be computationally universal just by itself! ...

MAT 200, Logic, Language and Proof, Fall 2015 Practice Questions

... n consecutive integers all of which are composite. Hint : Consider (n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + n + 1. Problem 8. Prove that there are infinitely many prime numbers which are congruent to 3 modulo 4. Hint : Proceed as in the proof of Theorem 23.5.1, but consider m = 4p1 p2 · · · pn ...

... n consecutive integers all of which are composite. Hint : Consider (n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + n + 1. Problem 8. Prove that there are infinitely many prime numbers which are congruent to 3 modulo 4. Hint : Proceed as in the proof of Theorem 23.5.1, but consider m = 4p1 p2 · · · pn ...

컴퓨터의 개념 및 실습 Practice 4 - Intelligent Data Systems Laboratory

... numbers are integer or not. The program should first prompt the user for how many numbers are to be entered. It should then input each of the numbers and print whether it is an integer number or not. * Integer does not mean the type of variable in this problem. For example, the program should consid ...

... numbers are integer or not. The program should first prompt the user for how many numbers are to be entered. It should then input each of the numbers and print whether it is an integer number or not. * Integer does not mean the type of variable in this problem. For example, the program should consid ...