Let L be the set of all languages over alphabet {0}. Show that L is uncountable,...

Let L be the set of all languages over alphabet {0}. Show that L is uncountable, using a proof by diagonalization.

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Colby Messinger answered 2 years ago

Given an alphabet Σ, what are the languages L over Σ for which L ∗ is...

Given an alphabet Σ, what are the languages L over Σ for which L ∗ is finite? How many such languages exist?

3. Are the following languages A and B over the alphabet Σ = {a, b, c,...

3. Are the following languages A and B over the alphabet Σ = {a, b, c, d} regular or nonregular? • For a language that is regular, give a regular expression that defines it. • For a nonregular language, using the pumping lemma prove that it is not regular. (a) A = {d 2j+1c k+1 | j ≥ k ≥ 0} · {c r+2b 2s+3 | r ≥ 0 and s ≥ 0} (b) B = {a 2j+2b k+3c j+1...

1. Let a < b. (a) Show that R[a, b] is uncountable

(10) Show that the Post Correspondence Problem is undecidable over the binary alphabet S = {0,...

(10) Show that the Post Correspondence Problem is undecidable over the binary alphabet S = {0, 1}.

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that...

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that X consists of 8 elements and that a metric d on X can be defined by ∀x,y ∈ X: d(x,y) := Number of places where x and y have different entries. 2: Show that the non-negativity of a metric can be deduced from only Axioms (M2), (M3), and (M4). 3: Let (X,d) be a metric space. Show that another metric D on X can...

1. Prove the language L is not regular, over the alphabet Σ = {a, b}. L...

1. Prove the language L is not regular, over the alphabet Σ = {a, b}. L = { aib2i : i > 0} 2) Prove the language M is not regular, over the alphabet Σ = {a, b}. M = { wwR : w is an element of Σ* i.e. w is any string, and wR means the string w written in reverse}. In other words, language M is even-length palindromes.

show that if I is uncountable,then 2I is not metrizable. hint: Suppose I as index set.

Let swap_every_two be an operation on languages that is defined as follows: swap_every_two(L) = {a2a1a4a3 ....

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. (it is a2a1a4a3 not a^2a^1 !!) 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 ≥...

Let U and V be vector spaces, and let L(V,U) be the set of all linear...

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Let L be the set of all linear transforms from R3 to R2 (a) Verify that...

Let L be the set of all linear transforms from R3 to R2 (a) Verify that L is a vector space. (b) Determine the dimension of L and give a basis for L.

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