Question

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 with respect to +, which is _________. Thus, 〈PD, +〉 is a

group!

3 Let D be the three-element set D = {a, b, c}. List the elements of PD. (For example, one element is {a},

another is {a, b}, and so on. Do not forget the empty set and the whole set D.) Then write the operation

table for 〈PD, +〉.

Answer #1

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 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 .
How many elements are there in P(D) ? Prove it!
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!
Which number is greater: the number of different subsets of
D which...

A subset of a power set.
(a)
Let X = {a, b, c, d}. What is { A: A ∈ P(X) and |A| = 2 }?
comment: Please give a clear explanation to what this
set builder notation translate to? Because I've checked the answer
for a) and it is A= {{a,b}, {a,c}, {a,d}, {b,c}, {b,d},
{c,d}}.
I don't understand because the
cardinality of A has to be 2 right? Meanwhile, the answer is
basically saying there's 6 elements. So...

Let A be a set with 20 elements.
a. Find the number of subsets of A.
b. Find the number of subsets of A having one or more
elements.
c. Find the number of subsets of A having exactly one
element.
d. Find the number of subsets of A having two or more elements.
(Hint: Use the answers to parts b and c.)

Using the following axioms:
a.) (x+y)+x = x +(y+x) for all x, y in R (associative law of
addition)
b.) x + y = y + x for all x, y elements of R (commutative law of
addition)
c.) There exists an additive identity 0 element of R (x+0 = x
for all x elements of R)
d.) Each x element of R has an additive inverse (an inverse with
respect to addition)
Prove the following theorems:
1.) The additive...

Let f and g be continuous functions from C to C and let D be a
dense
subset of C, i.e., the closure of D equals to C. Prove that if
f(z) = g(z) for
all x element of D, then f = g on C.

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).

Let G be a group
containing 6 elements a, b, c, d, e, and f. Under the group
operation called the multiplication, we know that ad=c, bd=f, and
f^2=bc=e. Which element is cf? How about af? Now find a^2. Justify
your answer.
Hint: Find the
identify first. Then figure out cb.

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∩Cc
c) ((A\D)∪B)∩(C ∩D)

Prove the statements (a) and (b) using a set element proof and
using only the definitions of the set operations (set equality,
subset, intersection, union, complement):
(a) Suppose that A ⊆ B. Then for every set C, C\B ⊆ C\A.
(b) For all sets A and B, it holds that A′ ∩(A∪B) = A′ ∩B.
(c) Now prove the statement from part (b)

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