Determine if the statement below is True or False. Justify your
answer by giving a proof or counterexample. Let A,B,C∈Mn×n(R) .
Suppose C is invertible and C=AB. Then the columns of A, B and C
are each bases for Rn and B is the change of basis matrix from the
columns of C to the columns of A.
Prove that all regular languages are context
free.
Note: Proof must proceed by
structural induction on regular
expressions
Please prove by Structural Induction. Will Upvote for
correct answer. Thanks
Determine whether or not the following languages are regular. If
the language is regular then give an NFA or regular expression for
the language. Otherwise, use the pumping lemma for regular
languages or closure properties to prove the language is not
regular.
1) L = { 0 n1 k : k ≤ n ≤ 2k}
2) L = { 0 n1 k : n > 0, k > 0 } È { 1 k0 n : k > 0, n...
Are the following languages over {a, b} regular? If they are
then prove it. If they are not prove it with the Pumping Lemma
a) {ap | p is a prime number}
b) {xax | x Î{a,b}*} (start by listing some strings
in, not in, the language
Prove that the following two statements are not logically
equivalent. In your proof, completely justify your answer.
(a) A real number is less than 1 only if its reciprocal is
greater than 1.
(b) Having a reciprocal greater than 1 is a sufficient condition
for a real number to be less than 1.
Proof:
#2.
Prove that the following is a valid argument:
All real numbers
have nonnegative squares.
The number i has a negative square.
Therefore, the...
Formal Languages
Give a regular expression for each of the following
languages:
L2a = {w ? {0,1}* | w corresponds to the binary encoding of
non-negative integers that are evenly divisible by 4
L2b = {w ? {a,b}* | w contains at least one 'a' and exactly two
b's}
L2c = {w ? {0, 1, 2}* | w starts with a 2, ends with a 1 and
contains an even number of 0's}.
Prove whether the following are regular (include the file) or
not regular (attach a proof). The alphabet is {0, 1}.
1. The set of strings that start with N 0's which are directly
followed by 2N 1's.
2. The set of strings start with two 0's, followed by N 1's,
followed by N 1's and end with three 0's.
3. The set of strings where every substring of length 5 has more
0's than 1's. Strings less than length five...
Do the following proofs deductively. Justify each step in your
proof with a law or inference rule.
a) If P ⇒ Q, ¬R ⇒¬Q, and P then prove R.
b) If P ⇒ (Q ∧ R) and ¬R ∧ Q then prove ¬P.
This project is intended to increase your understanding of
languages and grammars. In a regular grammar (RG), all
production rules must have one of the following forms:
Ni =
t1t2t3...tkNj
Ni =
t1t2t3...tk
where ti denotes a terminal (alphabet
symbol), and Nj and Nj denote
nonterminals. Any language defined by a regular grammar is
a regular language. Regular languages can also be defined by finite
automata, transition graphs, and regular expressions.
More complex grammars, such as context-free grammars
(CFG),...