Question

In: Computer Science

Problem 14.1.2 Let M be a DFA with transition function δ. Prove carefully, by induction for...

Problem 14.1.2 Let M be a DFA with transition function δ. Prove carefully, by induction for all strings w over M’s alphabet, that δ ∗ (i, w) = j if and only if there is a path from node i to node j in the graph of M such that the labels on the edges of the path, read in order, form w.

Expert Solution

Base case :- |w| = 1.

Let w= a.

Then

Clearly for |w|= 1, only if there is path (which is single edge in this case) in graph of M from i to j with label of edge being single letter w itself.

Hence the statement is true for base case.

Induction hypothesis :- if and only if there is path from i to j in the graph of M with labels on the edges of path in order form string w and |w| <= k.

Inductive step :- |w| = k+1

Let w = w'c where

Now if

Since |w'| = k. If then as per induction hypothesis , there must be path from node i to node k in graph of M with labels of edges being alphabets in w' from left to right.

Now,

This means that there is path from node i to node k and then there is an edge from node k to node j with label of edge being alphabet c. This means that there is path from node i to node j of length k+1.

Hence the argument that if and only if there is path from node i to node j in the graph of M satisfied even for |w|=k+1 which proves that given statement is correct for all string w.

Please comment for any clarification .

venereology answered 6 hours ago

Prove that the language L={(M, N): M is a Turing machine and N is a DFA...

Prove that the language L={(M, N): M is a Turing machine and N is a DFA with L(M) =L(N)} is undecidable. You need to derive a reduction from Atm={(M, w)|Turing machine M accepts w} to L. (In layman's terms please, no other theorems involved)

In this problem we prove that the Strong Induction Principle and Induction Principle are essentially equiv-...

In this problem we prove that the Strong Induction Principle and Induction Principle are essentially equiv- alent via Well-Ordering Principle. (a) Assume that (i) there is no positive integer less than 1, (ii) if n is a positive integer, there is no positive integer between n and n+1, and (iii) the Principle of Mathematical Induction is true. Prove the Well-Ordering Principle: If X is a nonempty set of positive integers, X contains a least element. (b) Assume the Well-Ordering Principle...

Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that...

Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the nth derivative of f(x).

Prove by Induction. Prop: if the (greatest common factor) gcf(a,m) = f then there is k,...

Prove by Induction. Prop: if the (greatest common factor) gcf(a,m) = f then there is k, l with ka + lm = f Steps that we have already established is, m = a(q1) + (r1) a = (r1)(q2) + (r2) (r1) = (r2)(q3) + (r3) ... r(n-2) = r(n+1)q(n) + r(n) r(n-1) = r(n)q(n+1) + 0 is the very last step to this sequence

Linear Algebra Carefully prove the following statement: Let A be an n×n matrix. Assume that there...

Linear Algebra Carefully prove the following statement: Let A be an n×n matrix. Assume that there exists an integer k ≥ 1 such that Ak = I . Prove that A is invertible.

Use double induction to prove that (m+ 1)^n> mn for all positive integers m; n

C. Prove the following claim, using proof by induction. Show your work. Let d be the...

C. Prove the following claim, using proof by induction. Show your work. Let d be the day you were born plus 7 (e.g., if you were born on March 24, d = 24 + 7). If a = 2d + 1 and b = d + 1, then an – b is divisible by d for all natural numbers n.

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Let N be a submodule of the R-module M. Prove that there is a bijection between...

Let N be a submodule of the R-module M. Prove that there is a bijection between the submodules of M that contain N and the submodules of M/N.

Let f be a one-to-one function from A into b with B countable. Prove that A...

Let f be a one-to-one function from A into b with B countable. Prove that A is countable. Section on Cardinality