Question

Appendix B Problem 3

Let Ω = {1,2,...,100} and let A,B and C be the following subsets of Ω.

A = {positive even numbers which are at most 100}

B = {two-digit numbers where the digit 5 appears}

C = {positive integer multiples of 3 which are at most 100}

D = {two-digit numbers such that the sum of the digits is 10}

List the elements of each of the following sets:

a) B\A

b) A∩B∩C^{c}

c) ((A\D)∪B)∩(C ∩D)

Answer #1

Let
U={1,2, 3, ...,3200}.
Let S be the subset of the numbers in U that are multiples of
4, and let T be the subset of U that are multiples of 9. Since
3200 divided by 4 equals it follows that
n(S)=n({4*1,4*2,...,4*800})=800
(a) Find n(T) using a method similar to the one that showed
that n(S)=800
(b) Find n(S∩T).
(c) Label the number of elements in each region of a two-loop
Venn diagram with the universe U and subsets S...

1. Let D={0,1,2,3,4,5,6,7,8,9} be the set of digits. Let P(D) be
the power set of D, i.e. the set of all subsets of D.
a) How many elements are there in P(D)? Prove
it!
b) Which number is greater: the number of different
subsets of D which contain the digit 7 or the number of different
subsets of D which do not contain the digit 7? Explain why!
c) Which number is greater: the number of different...

Let A be the last two digits, let B be the last digit, and let C
be the sum of the last three digits of your 8-digit
student ID. Example: for 20245347, A = 47, B = 7, and C = 14.
Consider two carts on a track. Cart A with mass mA =
(0.300 + (A/100)) kg has a speed of (2.50 + B) m/s before colliding
with and sticking to cart B. Cart has a mass mB =...

How to write a C++ program.
Additive persistence is a property of the sum of the digits of
an integer. The sum of the digits is found, and then the summation
of digits is performed creating a new sum. This process repeats
until a single integer digit is reached. Consider the following
example:
1. The beginning integer is 1234
2. Sum its digits is 1+2+3+4 = 10
3. The integer is now 10
4. The sum of its digits is...

question 1: Let A be the last digit, let B be the second to last
digit, and let C be the sum of the last three digits of your
8-digit student ID. Example: for 20245347, A = 7, B = 4, and C =
14.
Starting from rest, a student pulls a (25.0+A) kg box a distance
of (4.50 + B) m across the floor using a (134 + C) N force applied
at 35.0 degrees above horizontal. The coefficient...

1. a) Let f : C → D be a function. Prove that if C1
and C2 be two subsets of C, then
f(C1ꓴC2) = f(C1) ꓴ
f(C2).
b) Let f : C → D be a function. Let C1 and C2
be subsets of C. Give an example of
sets C, C1, C2 and D for which f(C ꓵ D) ≠
f(C1) ꓵ f(C2).

4. Let a = 24, b = 105 and c = 594.
(a) Find the prime factorization of a, b and c.
(b) Use (a) to calculate d(a), d(b) and d(c), where, for any
integer n, d(n) is the number of positive divisors of n;
(c) Use (a) to calculate σ(a), σ(b) and σ(c), where, for any
integer n, σ(n) is the sum of positive divisors of n;
(d) Give the list of positive divisors of a, b and c.

Let A,BA,B, and CC be sets such that |A|=11|A|=11, |B|=7|B|=7
and |C|=10|C|=10. For each element (x,y)∈A×A(x,y)∈A×A, we associate
with it a one-to-one function f(x,y):B→Cf(x,y):B→C. Prove that
there will be two distinct elements of A×AA×A whose associated
functions have the same range.

Let (a, b, c, d) be elements within N+ (all positive natural
numbers). If a is less than/equal to b, and c is less than/equal to
d, then (a*c) is less than/equal to (b*d) with equality if and only
if a=b and c=d.
Proof?

Thus, A + (B + C) = (A + B) + C.
If D is a set, then the power set of D is the set PD of all the
subsets of D. That is,
PD = {A: A ⊆ D}
The operation + is to be regarded as an operation on PD.
1 Prove that there is an identity element with respect to the
operation +, which is _________.
2 Prove every subset A of D has an inverse...

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