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Chapter 7: Q5E (page 322)

Is the following formula satisfiable?

(x∨y)∧(x∨y)∧(x∨y)∧(x∨y).

Short Answer

Expert verified

This techniqueXYXVYX-VYX-VY- satisfiable since it uses every one of the variables that produce true and pair values based on X and Y.

Step by step solution

01

Step 1:Satisfiable means

It implies that the level of XandYmay be expressed in terms of True or False. This algorithm must yield one actual worth in just about any mixture of true and false. So, let's set the true and false combinations in the XandYpositions.

02

Truth Table

True=TandFalse=FX-isthebalanceoftheX

Truth table approach:

XYXVYX-VYX-VY-FinalresultTTTTFFTFTFTFFTTTTTFFFFFF

This ultimate method, which is really the As well as operations of the all the parameters, provides the True value for a pair of X=FalsewithY=True, hence the formula is satisfiable. However, it is not a tautology because it gives the erroneous value for some values

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Most popular questions from this chapter

Show that if P=NP , a polynomial time algorithm exists that takes an undirected graph as input and finds a largest clique contained in that graph. (See the note in Problem 7.38.)

In the proof of the Cook–Levin theorem, a window is a 2×3rectangle of cells. Show why the proof would have failed if we had used role="math" localid="1664195743361" 2×2windows instead.

A cut in an undirected graph is a separation of the vertices V into two disjoint subsets S and T . The size of a cut is the number of edges that have one endpoint in S and the other in T . Let MAX-CUT=<G,K>|Ghasacutofsizekormore.

Show that MAX-CUT is NP-complete. You may assume the result of Problem 7.26. (Hint: Show that≠SAT⩽PMAXCUT. The variable gadget for variable x is a collection of 3c nodes labeled with x and another nodes labeled with x . The clause gadget is a triangle of three edges connecting three nodes labeled with the literals appearing in the clause. Do not use the same node in more than one clause gadget. Prove that this reduction works.)

Let G represent an undirected graph. Also let

SPATH={́aG,a,b,k~n∣G contains a simple path of length atmost k from a to b},and

LPATH={́aG,a,b,k~n∣G contains a simple path of length atleast k from a to b}.

a) Show that SPATH? P.

b) Show that LPATH is NP-complete.

You are given a box and a collection of cards as indicated in the following figure. Because of the pegs in the box and the notches in the cards, each card will fit in the box in either of two ways. Each card contains two columns of holes, some of which may not be punched out. The puzzle is solved by placing all the cards in the box so as to completely cover the bottom of the box (i.e., every hole position is blocked by at least one card that has no hole there). It represents a card and this collection of cards has a solution}. Show that PUZZLE is NP-complete.
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