Question

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 be
countable. **HINT**: Try to argue that it is possible
to arrange the elements of S into an infinite sequence.

b. Now assume that we have proved that S is indeed countable.
Use this fact to prove (formally, this time) that the following
set, denoted by T, is also countable: T = {X ⊆ N| |X| is finite and
is a prime number}. **HINT**: You might want to first
argue that the set {X ⊆ N| |X| is finite} is countable.

Answer #1

Let X be a topological space with topology T = P(X). Prove that
X is finite if and only if X is compact. (Note: You may assume you
proved that if ∣X∣ = n, then ∣P(X)∣ = 2 n in homework 2, problem 2
and simply reference this. Hint: Ô⇒ follows from the fact that if X
is finite, T is also finite (why?). Therefore every open cover is
already finite. For the reverse direction, consider the
contrapositive. Suppose X...

Let B be the set of
all binary strings of length 2; i.e. B={ (0,0), (0,1), (1,0),
(1,1)}. Define the addition and multiplication as coordinate-wise
addition and multiplication modulo 2. It turns out that B becomes a
Boolean algebra under those two operations. Show that B under
addition is a group but B under multiplication is not a group.
Coordinate-wise
addition and multiplication modulo 2 means (a,b)+(c,d)=(a+c, b+d),
(a,b)(c,d)=(ac, bd), in addition to the fact that 1+1=0.

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