Question

Let n be an odd number and X be a set with n elements. Find the no. of subsets of X which has even no. of elements. The answer should be a number depending only on n. (For example, when n = 5, you need to find the no.of subsets of X which has either 0 elements or 2 elements or 4 elements)

Answer #1

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.)

Suppose that the set S has n elements and discuss the
number of subsets of various sizes.
(a) How many subsets of size 0 does S have?
(b) How many subsets of size 1 does S have?
(c) How many subsets of size 2 does S have?
(d) How many subsets of size n does S have?
(e) Clearly the total number of subsets of S must equal the sum of
the number of subsets of size 0, of size...

Consider the set S = {x1, x2, . . . ,
x2} of n distinct elements. Find the number of
even-sized subsets of S? For example, if S = {a, b, c}, then there
are four even-sized subsets of S – {}, {a, b}, {a, c}, and {b, c}.
Justify your answer.

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Let N be a positive integer random variable with PMF of the
form
pN(n)=1/2⋅n⋅2^(−n),n=1,2,….
Once we see the numerical value of N, we then draw a random
variable K whose (conditional) PMF is uniform on the set
{1,2,…,2n}.
Find the marginal PMF pK(k) as a function of k. For simplicity,
provide the answer only for the case when k
is an even number. (The formula for when k is odd
would be slightly different, and you do not need to...

Let X be a set and let (An)n∈N be a sequence of subsets of X.
Show that: (a) If (An)n∈N is increasing, then liminf An = limsupAn
=S∞ n=1 An. (b) If (An)n∈N is decreasing, then liminf An = limsupAn
=T∞ n=1 An.

A)Let the Universal Set, S, have 118 elements. A and B are
subsets of S. Set A contains 18 elements and Set B contains 94
elements. If the total number of elements in either A or B is 95,
how many elements are in B but not in A?
B)A company estimates that 0.3% of their products will fail
after the original warranty period but within 2 years of the
purchase, with a replacement cost of $350.
If they offer...

3. Prove by contrapositive: Let n ∈ N. If n^3−5n−10>0,then n
≥ 3.
4. Prove: Letx∈Z. Then5x−11 is even if and only if x is odd.
4. Prove: Letx∈Z. Then 5x−11 is even if and only if x is
odd.

Let SS be the universal set, where:
S={1,2,3,...,28,29,30}S={1,2,3,...,28,29,30}
Let sets AA and BB be subsets of SS, where:
Set
A={1,8,13,14,16,17,20,25,27,28}A={1,8,13,14,16,17,20,25,27,28}
Set B={6,7,9,13,14,30}B={6,7,9,13,14,30}
Set
C={5,8,10,11,13,14,15,17,23,25,28}C={5,8,10,11,13,14,15,17,23,25,28}
Find the number of elements in the set (A∩B)(A∩B)
n(A∩B)n(A∩B) =
Find the number of elements in the set (B∩C)(B∩C)
n(B∩C)n(B∩C) =
Find the number of elements in the set (A∩C)(A∩C)
n(A∩C)n(A∩C) =

Let N denote the set of positive integers, and let x be a number
which does not belong to N. Give an explicit bijection f : N ∪ x →
N.

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