By constructing a suitable bijection, show that the number of
subsets of an n-set of odd...
By constructing a suitable bijection, show that the number of
subsets of an n-set of odd size is equal to the number of subsets
of an n-set of even size.
Let S be a set of n numbers. Let X be
the set of all subsets of S of size k, and let
Y be the set of all ordered k-tuples
(s1, s2, ,
sk)
such that
s1 < s2
< < sk.
That is,
X
=
{{s1, s2, ,
sk} | si S and all si's
are distinct}, and
Y
=
{(s1, s2, ,
sk) | si S and s1 <
s2 < < sk}.
(a) Define a one-to-one correspondence
f : X → Y.
Explain...
Let S(n) be the number of subsets of {1,2,...,n} having the
following property: there are no three elements in the subset that
are consecutive integers. Find a recurrence for S(n) and explain in
words why S(n) satisfies this recurrence
6.3.8. Problem. Let f : A → B be a continuous bijection
between subsets of R.
(a) Show by example that f need not be a homeomorphism.
(b) Show that if A is compact, then f must be a
homeomorphism.
6.3.9. Problem. Find in Q a set which is both relatively
closed and bounded but which is not compact.
Prove the following statements!
1. There is a bijection from the positive odd numbers to the
integers divisible by 3.
2. There is an injection f : Q→N.
3. If f : N→R is a function, then it is not surjective.
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to find and confirm them.Show that the number of nodes of both
the fixed-tuple...
Let P(n) := ” If n^3 is odd then n is also odd.” I.e., if ∃k ∈
Z, n3 = 2k + 1, ∃b ∈ Z, n = 2b + 1
a) Prove P(n) by contraposition
b) Prove P(n) contradiction
c) Prove P(n) using induction
(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable.
(2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable.
(3) Show that the intersection of two countable sets is
countable.
(4) Show that the set of all irrational numbers is
uncountable.
(5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · ·
is...