Question

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.

Answer #1

Using the completeness axiom, show that every nonempty set E of
real numbers that is bounded below has a greatest lower bound
(i.e., inf E exists and is a real number).

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

Counting theory: Find how many 4-digit positive
integers are there with no repeating digits (e.g.: 5823) or where
digit repetition is allowed but all digits must be odd (e.g.: 5531
satisfies this condition but 7726 and 6695 do not since they
contain even digits).

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)

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.

Use the method of direct proof to show that for any positive
5-digit integer n, if n is divisible by 9, then some of its digits
is divisible by 9 too.

discrete math (3) with full proof
Use the Well Ordering principle to show that a set S of positive
integers includes 1 and which includes n+ 1, whenever it includes
n, includes every positive integer.

Use
mathematical induction to show that ?! ≥ 3? + 5? for all integers ?
≥ 7.

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

Use a proof by induction to show that, −(16−11?) is a positive
number that is divisible by 5 when ? ≥ 2.
Prove (using a formal proof technique) that any sequence that
begins with the first four integers 12, 6, 4, is neither arithmetic
nor geometric.

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