(1) Show that the set { 1 m + 1 n : m, n ∈ N}...

(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable.

(2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable.

(3) Show that the intersection of two countable sets is countable.

(4) Show that the set of all irrational numbers is uncountable.

(5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · · is uncountable. [Hint: Imitate the proof we had for E × E × · · ·, where E = {0, 1}.]

Expert Solution

Colby Messinger answered 5 months ago

There are (m − 1)n + 1 people in a room. Show that either there are...

There are (m − 1)n + 1 people in a room. Show that either there are m people who mutually do not know each other, or there is a person who knows at least n others.

Show that M is a subgroup N; N is a subgroup D4, but that M is not a subgroup of D4

D4 = {(1),(1, 2, 3, 4),(1, 3)(2, 4),(1, 4, 3, 2),(1, 2)(3, 4),(1, 4)(2, 3),(2, 4),(1, 3)} M = {(1),(1, 4)(2, 3)} N = {(1),(1, 4)(2, 3),(1, 3)(2, 4),(1, 2)(3, 4)} Show that M is a subgroup N; N is a subgroup D4, but that M is not a subgroup of D4

1. Must be nicely written up AS A PROOF. a. Show that gcd(m + n, m)...

1. Must be nicely written up AS A PROOF. a. Show that gcd(m + n, m) = gcd(m, n). b. If n | k(n + 1), show that n | k. c. Show that any two consecutive odd integers are relatively prime.

Show that |N| = |Z|, where N is the set of natural numbers and Z is...

Show that |N| = |Z|, where N is the set of natural numbers and Z is the set of all integers.

Let A be a set with m elements and B a set of n elements, where...

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

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....

Prove via induction the following properties of Pascal’s Triangle: •P(n,2)=(n(n-1))/2 • P(n+m+1,n) = P(n+m,n)+P(n+m−1,n−1)+P(n+m−2,n−2)+···+P(m,0) for all...

Prove via induction the following properties of Pascal’s Triangle: •P(n,2)=(n(n-1))/2 • P(n+m+1,n) = P(n+m,n)+P(n+m−1,n−1)+P(n+m−2,n−2)+···+P(m,0) for all m ≥ 0

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(1) Show that the set { 1 m + 1 n : m, n ∈ N}...

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