Question

Show that the set of all functions from the positive integers to the set {1, 2, 3} is uncountable.

Answer #1

1.) Given any set of 53 integers, show that there are two of
them having the property that either their sum or their difference
is evenly divisible by 103.
2.) Let 2^N denote the set of all infinite sequences consisting
entirely of 0’s and 1’s. Prove this set is uncountable.

1. A) Show that the set of all m by n matrices of integers is
countable where m,n ≥ 1 are some ﬁxed positive integers.

Show that for all positive integers n
∑(from i=0 to n) 2^i=2^(n+1)−1
please use induction only

Show that the set of all integers that are multiplies of 3 is a
commutative ring. Does it have an identity?

Use Axiom of Completeness to show that the set of positive
integers that contain digit 7 in their decimal expansion (for
example, 47, 1976 or 172760) is unbounded.

Prove that for all positive integers n,
(1^3) + (2^3) + ... + (n^3) = (1+2+...+n)^2

Remember that the DOMAIN is INTEGERS
(....,-2,-1,0,1,2,3,4.......) and the TARGET is POSITIVE INTEGERS
(1,2,3,4,.....) .
Give explanation and proofing for each .
Find a function whose domain is the set of all
integers and whose target is the set of all
positive integers that satisfies each set of
properties.
(a) Neither one-to-one, nor onto.
(b) One-to-one, but not onto.
(c) Onto, but not one-to-one.
(d) One-to-one and onto.

Show that n = ∑ d ∣ n ϕ ( d ) for all positive integers n.

4. Let Z be the set of all integers (positive, negative and
zero.) Write a
sequence containing every element of Z.

a) Show that the set Q+ of positive rationals and set Q− of
negative rationals are equivalent.
b Show that the set of even integers and set of odd integers are
equivalent.
Please answer questions in clear hand-writing and show me the
full process, thank you. (Sometimes I get the answer which was
difficult to read)

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