In: Advanced Math

a. What are the maximal and minimal elements, if any, of the
set (N+,|)? Is there a minimum or maximum element?
(N+={1,2,3,4,...}).

b. There are five flavors of icecream: banana, chocolate,
lemon, strawberry and vanilla. We can have three scoops. How many
variations will there be?

a.

N = {1, 2, 3,...}

The minimal element of this set is 1. But since this is unbounded above so maxiy element don't exist.

b.

If all 3 scoops have the same flavour the there are
5C_{1} = 5 possibilities.(as there are five flavors of ice
cream)

If 2 scoops with the same flavour and 1 with different flavor of
ice cream then there are 2× 5C_{2} = 20 = 5×4
possibilities.( For if two scoop is filled with one type of ice
cream then third schoop can be filled by 4 flavors, there are 5
ways in which two scoops can be filled by same type of ice cream as
there are 5 types of ice cream so 5×4) or (first find how many ways
you can choose 2 flavors: 5C_{2}. Then remember that there
are 2 ways to have 3 scoops of 2 flavors, such as with chocolate
and vanilla. 2 scoops vanilla, 1 chocolate, or 2 chocolate, 1
vanilla)

If all 3 scoops have different flavour of ice cream then there
are 5C_{3} = 10 possibilities.

So there are total 5 + 20 + 10 = 35 variations.

What is the minimal sample size needed for a 95% confidence
interval to have a maximal margin of error of 0.1 in the following
scenarios? (Round your answers up the nearest whole number.)
(a) a preliminary estimate for p is 0.16
(b) there is no preliminary estimate for p

What is the minimal sample size needed for a 95% confidence
interval to have a maximal margin of error of 0.1 in the following
scenarios? (Round your answers up the nearest whole number.)
(a) a preliminary estimate for p is 0.28
(b) there is no preliminary estimate for p

What is the minimal sample size needed for a 95% confidence
interval to have a maximal margin of error of 0.1 in the following
scenarios? (Round your answers up the nearest whole number.)
(a) a preliminary estimate for p is 0.39
(b) there is no preliminary estimate for p

what is the minimal sample size needed for a 95% confidence
interval to have a maximal margin of error of 0.1 in the following
scenarios? ( Round your answers up to the nearest whole number)
(a) a preliminary estimate for p is 0.34
(b) there is no preliminary estimate for p

What is the minimal sample size needed for a 95% confidence
interval to have a maximal margin of error of 0.1 in the following
scenarios? (Round your answers up the nearest whole number.)
(a) a preliminary estimate for p is 0.31
(b) there is no preliminary estimate for p
For this problem, carry at least four digits after the decimal
in your calculations. Answers may vary slightly due to
rounding.
A random sample of medical files is used to estimate...

What is the minimal sample size needed for a 95% confidence
interval to have a maximal margin of error of 0.1 in the following
scenarios? (Round your answers up the nearest whole number.)
(a) a preliminary estimate for p is 0.15
(b) there is no preliminary estimate for p

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.

(Lower bound for searching algorithms) Prove: any
comparison-based searching algorithm on a set of n elements takes
time Ω(log n) in the worst case. (Hint: you may want to read
Section 8.1 of the textbook for related terminologies and
techniques.)

Show that {t_(1,s) : 2 ≤ s ≤ n} is a minimal generating set for
S_n. You may use the fact that {t_(r,s) : 1 ≤ r < s ≤ n}, as
defined in the outline, generates S_n.

Let f(n,k) be the number of equivalence relations with k classes
on set with n elements.
a) What is f(2,4)?
b) what is f(4,2)?
c) Give a combinational proof that f(n,k) = f(n-1,k-1)+k *
f(n-1,k)

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