We learnt in class/Notes that A[x] → (Vx)A[x] is NOT a theorem if x indeed occurs free in A. Yet, I think I got a proof: First, let me use DThm and prove instead A[x] F (Vx)A[x]: 1) A[x] 2) (Vz)A[r] (1+ Gen) (hyp) 2.

Q: Proof using laws of logic I) (paq) → p is tautology II) -p+q p +-

A: I) (p ∧ q) → p = ¬(p ∧ q) ∨ p  [Expanding → ] = ¬p ∨ ¬p ∨ p [Expanding negation] = ¬p ∨ p   = True…

Q: Suppose g : A → A and f : A → A, where A = {1,2,3,4}, and the functions f and g are defined by f =…

A:

Q: Write the ô function of the following (1) dfa (º DFA:QX2→Q) and (2) nf SNFA:Q×(EU{1})→P(Q) )…

A: DFA : DFA stands for deterministic finite automata. This machine (DFA) reads an input string one…

Q: 3.3. Consider the cascade combination of two systems shown in Fig. P.3.3(a). G.G w/n] Sys, { } Sys,{…

A: Given The answer is given below.

Q: Identify the scope of each of the quantifiers in the following wffs and indicate any free variables.…

A: The formula known as a "Well-Formed Formula" (WFF) consists of variables (capital letters),…

Q: Let L be the language {zzk : z belongs to E*}. Use the Pumping Lemma to show that L is not regular,…

A: Input Language : L = {zzk : z belongs to {m,n}* ,  k is any integer} Expanding Language : Using…

Q: Construct a truth table to show if (¬p V g) + (¬p A q) is a tautology or a contradiction or neither.…

A: A pattern obtained by combining two or more statements using logical connectives is called statement…

Q: Is the wff valid or invalid? Explain.(∀x)[P(x) ∨ Q(x)]  → (∀x)P(x) ∨ (∀x)Q(i)

A: Here below we use predicate logic rules to check it is valid or not?…

Q: :Q×(EU{2})→ P(Q)) respectively, where E={a,b}. а, b a a,b a,b b a

A: 1)  Let the final state be called q2 and other remaining state be q1.  Therefore the transition…

Q: Given: ∀x [p(x) → q(x)] ∀x[p(x) → (∀y w(y))] ∀x ∀y [(q(x) ∧ w(y)) → s(x)] p(MARY) Show s(MARY)…

A: The answer is as follows.

Q: 2) Is aa distinguishable from bb with respect to the set of strings over {a,b} that have an even…

A: ^b*(ab*ab*)*$^a*ba*(ba*ba*)*$The first regular expression ensures  there are an even number of a…

Q: (p → (q v r)) → ((p ^ ¬q) → r) is O Contingency O Tautology O None of the choices O Contradiction O…

A: understanding some basic terms  1) tautology - a tautology is a proposition that is always true  2)…

Q: Show that [¬P∧ (P ∨ Q) ]→Q is tautology using laws of equivalences

A: Given logic is,  [¬P∧ (P ∨ Q) ]→Q Variables present in this function are P, Q.

Q: Properties of U, 12, 6 Let f, g, h: N →R*. Then, P1: (Transitivity) fe O(g) and g e O(h)=fe 0(h) How…

A: Transitivity of all notation: (n) = Ο(g(n) ) and g(n) = Ο(h(n) ) ≈ f(n) = Ο(h(n) ) f(n) = Ω(g(n) )…

Q: Algorithm for B-ssgs using Homomorphism Input: soluble permutation groups G and H; a base B for G; a…

A: Algorithm for B-ssgs using Homomorphism given data:-Input: soluble permutation groups G and H;a base…

Q: Prove that (a + b) mod m = [(a mod m) + (b mod m)] mod m

A: The problem asks for a proof of a modular arithmetic property:Given: and (which means is a…

Q: EXERCISE 5.3 Verify whether (a → b) → b is a tautology. Is the following proof correct? If not, what…

A: A tautology is a formula that is always true, regardless of the truth values of its propositional…

Q: Let A E R³x7 and b E R³ be given by 3 9 -2 -7 6. -7 -5 7 A = 4 -10 4 -10 4 and b = -2 -10 -8 -4 -1 6…

A:

Q: Prove that each of the following is a valid argument: (vX)P(x) ^ (3x)Q(x) → (3x)[P(x) ^ Q(x)]…

A: The notation (∀x)P(x) denotes the universal quantification of P(x). Here ∀ is called the universal…

Q: A = {[(¬P → Q) AR] V ¬Q} → (QAR)

A: Answer: Answer is given in step 2

Q: - universal generalization (twice-applied) that (((C − A) n B) u (D n B)) = (B n ((C − A) U D)).

A: Answers:- To prove that (((C - A) n B) U (D n B)) = (B n ((C – A) U D)) using universal…

Q: Transform the ER model below into a set of 3NF relations. Use the format shown here for relations:…

A: A database is a collection of related data which represents some aspect of the real world. A…

Q: Show that the number of as that appear in a lengthn (c) string in {a, b, c} that doesn't have aa as…

A: Recurrence Relation: “ In science, a repeat connection is an condition that recursively…

Q: 2. We learnt in class/Notes that A[x] → (Vx)A[x] is NOT a theorem if x indeed occurs free in A. Yet,…

A: The solution to the given question is:

Q: It was claimed that: (a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N.…

A: A well-ordering is a total order on a set where every non-empty subset has at least element. It must…

Q: Is the following a CFL? If yes then provide a pushdown automaton for it (no need to prove the…

A: The solution to the given question is:

Q: Prove that all elements in S are multiples of 3 by structural induction • 3 € S x, y ES → (x + y) E…

A: Structural Induction : Proving something about a recursively defined object using an inductive proof…

Q: Minimize using K- map the logical function F(x;Y,z) = SOP C1, 3,5,7) ?

A: Given : Minterms = 1,3,5,7

Q: (a) Show that 3x(Px x Px). (b) Show that {Qx,V y(Qy → Pz)} EX Px.

A: The solution for the above-given question is given below:

Q: Prove (using the concept of interpretations and the value of a formula under v1 an interpretation I)…

A: Below is the answer to above question. I hope this will be helpful for you...

Q: Prove that                                [(∀x)A(x)]′ ↔ (∃x)[A(x)]′is valid.

A: This is a tautology and we have to prove it by contradiction. Suppose, A(x) (domain) is D and the…

Q: 3. Let F be a formula where no atomic formula is repeated. Give a procedure that finds a smallest…

A: Answer is given below in detail and correct way

Q: K- map the logical Minimize function F(x;y, z)=SOP C1,3,5,7) ? using

A: Given Data : Minterms = 1 , 3 , 5 , 7

Q: lek A=B = {E1,2,3,47 and R={Lasb)! a=b} wril.e MR and show that its equivalent of 1. not 2 show by…

A:

Q: Prove f(n) = n3 - (n2 log2 n) + 2n f(n) ∈ O(n3) by giving the constants (c, n0) and arguing that…

A: In this question, it is asked to prove that the f(n) ∈ O(n3) if f(n) = n3 - (n2 log2 n) + 2n It is…

Q: Prove that the following instance of PCP does not have a match [aba, bb], [ab, a], [bab, aa], [ab,…

A: Here we have given solution for PCP that does not have a match [aba, bb], [ab, a], [bab, aa], [ab,…

Q: Prove Pierce's law: For any propositions P and Q, the proposition ((P -+ Q) -> P) -+ P is a…

A: Proof is given below in detail

Q: Given interpretations (a) (Vx) (y) P(x, y) → (3x)(Vy)P(x, y) (b) (Vx) [P(x) → Q(x)] → [(3x)P(x) →…

A: A. (∀x)( ∃y)P(x,y) → (∃x)( ∀y)P(x,y) This wff is invalid because the quantifiers cannot be changed.…

Q: 2.Use the method of proof by contradiction to show that [(pvg)^(p→r)^(q→r)]→r r is a tautology.

A: Use the method of proof by contradiction to show that  [(p∨q)∧(p→r)∧(q→r)]→r is  a tautology

Q: 2. Prove that [(pA (~r)) V (r^(-q))] V (~p V q) E F using the direct proof approach (axiomatic…

A: //note : axiomatic method :uses only basic axioms and rules of boolean algebra to prove given:…

Eight puzzle problem

The N-puzzle problem contains N tiles (N+1 including empty tiles) where the value of N can be 8, 15, 24, etc. The puzzle is divided into the square root of (N+1) rows and the square root of (N+1) columns. These problems have an initial state or configuration and a goal state or configuration. The puzzle can be solved by shifting the tiles one by one in the empty space. The objective is to place the numbers on the tiles to match the final state using the empty space. The other name for the N-puzzle problem is the sliding puzzle.