8. Show that the set of integer numbers is countable (hint: find a one to one...

8. Show that the set of integer numbers is countable (hint: find a one to one mapping with the set of natural numbers which is countable by definition).

Expert Solution

Colby Messinger answered 2 years ago

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the...

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the statement that 13 + 23 +· · ·+n3 =(n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you use the inductive hypothesis....

(11) Prove that a union of two countable sets is countable. (Hint: the same idea used...

(11) Prove that a union of two countable sets is countable. (Hint: the same idea used to show that Z is countable might be useful.) (Don’t forget that countable sets can be finite.) (12) We saw in class that N × N ∼ N is countable. Prove that A × B is is countable for any countable sets A, B. (Hint: If you can prove that A × B ∼ N × N then you can use what has already...

Use cardinality to show that between any two rational numbers there is an irrational number. Hint:...

Use cardinality to show that between any two rational numbers there is an irrational number. Hint: Given rational numbers a < b, first show that [a, b] is uncountable. Now use a proof by contradiction

show that if I is uncountable,then 2I is not metrizable. hint: Suppose I as index set.

1. (a) Explain what is meant by the terms countable set and uncountable set. Give a...

1. (a) Explain what is meant by the terms countable set and uncountable set. Give a concrete example of each. b) Show that if A and B are countably infinite then the set A x B is also countably infinite. c) Give an expression for the cardinality of set A U B when A and B are both finite sets. d) What can you say about the cardinality of the set A U B when A and B are infinite...

8. Definition: A set A is finite if there exists a non-negative integer c such that...

8. Definition: A set A is finite if there exists a non-negative integer c such that there exists a bijection from A to {n ∈ N : n ≤ c}. (The integer c is called the cardinality of A.) (a) Let A be a finite set, and let B be a subset of A. Prove that B is finite. (Hint: induction on |A|. Note that our proof can’t use induction on |B|, or indeed refer to “the number of elements...

Consider the set of integer numbers from 0 to 9, that is {0, 1, 2, 3,...

Consider the set of integer numbers from 0 to 9, that is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Bob wishes to use these numbers to create a 7-digit password to secure his new laptop. Note that each number can appear in any position (for example, 0 can be the first number in the password). (a) Find the number of 7-digit passwords that are possible. (b) Find the number of 7-digit passwords with distinct digits. (c) Find...

Show that the set {DiracDelta(t-tao): t, tao are real numbers} is a basis set for the...

Show that the set {DiracDelta(t-tao): t, tao are real numbers} is a basis set for the analog complex signal space.

Find a primitive root, for all positive integral m, modulo each integer below. (a) 7m (Hint:...

Find a primitive root, for all positive integral m, modulo each integer below. (a) 7m (Hint: Using Corollary 5.15, find a common primitive root r modulo 7 and 72• The proof of Proposition 5.17 then guarantees that r is a primitive root modulo 72 for all positive integral m.) (b) 11m (c) 13m (d) 17m

Write a program to find the prime numbers IN JAVA Ask user to input the integer...

Write a program to find the prime numbers IN JAVA Ask user to input the integer number test the number whether it is a prime number or not Then, print “true” or “false” depending on whether the number is prime or isn’t. Hint: number is prime when is has exactly 2 factors: one and itself. By this definition, number 1 is a special case and is NOT a prime. Use idea of user input, cumulative sum, and loop to solve...

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