Question

For a set A, let P(A) be the set of all subsets of A. Prove that A is not equivalent to P(A)

Answer #1

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.

6. 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 ⊆ B.
(a) Is this relation reflexive? Explain your
reasoning.
(b) Is this relation symmetric? Explain your
reasoning.
(c) Is this relation transitive? Explain your
reasoning.

Let {??}?∈? be an indexed collection of subsets of a set ?.
Prove:
a. ?\(⋃ ??) ?∈? = ⋂ (?\??) ?∈?
b. ?\(⋂ ??) = ⋃ (?\??) ?∈?? ∈?
Note: These are DeMorgan’s Laws for indexed collections of
sets.

Discrete mathematics function relation
problem
Let P ∗ (N) be the set of all nonempty subsets of N. Define m :
P ∗ (N) → N by m(A) = the smallest member of A. So for example, m
{3, 5, 10} = 3 and m {n | n is prime } = 2.
(a) Prove that m is not one-to-one.
(b) Prove that m is onto.

Prove that the set of all finite subsets of Q is countable

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...

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 set of all natural numbers less than 100.
How many subsets with three elements does set A have such that the
sum of the elements in the subset must be divisible by 3?

Is
it true that for all subsets A and B of a set U, there is a subset
X of U for which A△X⊆B△X? If there is such an X, then prove it (in
particular, say what X can be, and prove your assertion); if there
can fail to be such an X, then give an example where there is no
such X.
Write legibly and do not skip steps.

Prove or disprove:
If A and B are subsets of a universal set U such that A is not a
subset of B and B is not a subset of A, then A complement is not a
subset of B complement and B complement is not a subset of A
complement

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