Question

Let F be the set of all finite languages over alphabet {0, 1}. Show that F is countable

Answer #1

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Q2 [10 pts] Give DFA's accepting the following languages
over the alphabet {0,1}:
a) The set of all strings whose 3rd symbol from the right end
is a 0.
b) The set of strings such that the number of 0's is divisible
by 3 and the number of 1's divisible by 2.

Which one of the following languages over the alphabet {0,1} is
described by the regular expression
(0+1)* 0 (0+1)* 0 (0+1)* ?
a.The set of all strings that begin and end with either 0 or
1
b.The set of all strings containing at most two zeros
c.The set of all strings containing at least two zeros.
d.The set of all strings containing the substring 00

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 finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

Let F = {A ⊆ Z : |A| < ∞} be the set of all finite sets of
integers. Let R be the relation on F defined by A R B if and only
if |A| = |B|. (a) Prove or disprove: R is reflexive. (b) Prove or
disprove: R is irreflexive. (c) Prove or disprove: R is symmetric.
(d) Prove or disprove: R is antisymmetric. (e) Prove or disprove: R
is transitive. (f) Is R an equivalence relation? Is...

Please answer True or False on the following:
1. Let L be a set of strings over the alphabet Σ = { a, b }. If
L is infinite, then L* must be infinite (L* is the Kleene closure
of L)
2. Let L be a set of strings over the alphabet Σ = { a, b }. Let
! L denote the complement of L. If L is finite, then ! L must be
infinite.
3. Let L be...

Let swap_every_two be an operation on languages that is defined
as follows:
swap_every_two(L) = {a2a1a4a3 . . . a2na2n−1 | a1a2a3a4 . . .
a2n−1a2n ∈ L where a1, . . . , a2n ∈ Σ} In this definition, Σ is
the alphabet for the language L.
1. What languages result from applying swap every two to the
following languages:
(a) {1 n | n ≥ 0}, where the alphabet is {1}.
(b) {(01)n | n ≥ 0}, where the...

Let A be a finite set and f a function from A to A.
Prove That f is one-to-one if and only if f is onto.

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

Let V and W be finite-dimensional vector spaces over F, and let
φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V )
= n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can
be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn},
for some vectors vk+1, . . . , vn ∈ V . Prove that...

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