Question

- 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 contain more than five digits from*D*(like {0,1,2,5,7,9} or*D*itself) or the number of different subsets of*D*which contain less than five digits from*D*(like {3,4,6,8} or*∅*)? Explain why!

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 S be a finite set and let P(S) denote the set of all subsets
of S. Define a relation on P(S) by declaring that two subsets A and
B are related if A and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

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...

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...

Answer the following brief question:
(1) Given a set X the power set P(X) is ...
(2) Let X, Y be two infinite sets. Suppose there exists an
injective map f : X → Y but no surjective map X → Y . What can one
say about the cardinalities card(X) and card(Y ) ?
(3) How many subsets of cardinality 7 are there in a set of
cardinality 10 ?
(4) How many functions are there from X =...

Given a random permutation of the elements of the set
{a,b,c,d,e}, let X equal the number of elements that are in their
original position (as listed). The moment generating function is X
is: M(t) = 44/120 + 45/120e^t + 20/120e^2t + 10/120e^3t+1/120e^5t
Explain Why there is not (e^4t) term in the moment generating
function of X ?

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

Let A be a nonempty set and let P(x) and Q(x) be open
statements. Consider the two statements (i) ∀x ∈ A, [P(x)∨Q(x)] and
(ii) [∀x ∈ A, P(x)]∨[∀x ∈ A, Q(x)]. Argue whether (i) and (ii) are
(logically) equivalent or not. (Can you explain your answer
mathematically and by giving examples in plain language ? In the
latter, for example, A = {all the CU students}, P(x) : x has last
name starting with a, b, ..., or h,...

Activity 10.5.
Suppose A is a set that definitely does not contain any cats,
and let
f:P(A)→P(A∪{Grumpy Cat})
represent the function defined by
f(X)=X∪{Grumpy Cat}
(a) Verify that f is injective.
(b)Verify that f is not surjective.
(c) Describe specifically how to restrict the codomain of f to
make it bijective.
restricting the codomain
the “induced” function X→B created from function f:X→Y and
subset B⊆Y by “forgetting” about all elements of Y that do not lie
in B, where B...

Activity 10.5.
Suppose A is a set that definitely does not contain any cats,
and let
f:P(A)→P(A∪{Grumpy Cat})
represent the function defined by
f(X)=X∪{Grumpy Cat}
(a) Verify that f is injective.
(b)Verify that f is not surjective.
(c) Describe specifically how to restrict the codomain of f to
make it bijective.
restricting the codomain
the “induced” function X→B created from function f:X→Y and
subset B⊆Y by “forgetting” about all elements of Y that do not lie
in B, where B...

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