Discrete Mathematics and Its Applications with MathZone

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
ICS 6D ( Discrete Mathematic): Solution to Homework #1 Section 1.1, problem 62 Let’s denote: K = “Kevin is chatting” A = “Abby is chattnig” H = “Heather is chatting” R = “Randy is chattnig” V = “Vijay is chatting” Here’s what we know about the world (in the order as they are stated in this exercise): 1. K H 2. R V 3. A R 4. ( V K ) ( ¬ V ∧ ¬ K ) 5. H ( A K ) So how do we know which combination is true? We have five variables here and 5 logical formulas which impose relations between these variables. So what we can do is to do a series of transformations which would simplify these relations: 2’. We can replace (2) by equation R = ¬ V . 3’. Therefore we can replace (3) by A → ¬ V . 4’. Also we can replace (4) by an equivalent expression V K Now let’s pick some variable on which lots of other variables seem to depend. For example pick H , set it to true, and see what happens to the other variables: If H then by (5) we immediately get A and K . (1) is true. By (3’) from A we get ¬ V . However, by (4’)
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: From ¬ V we get ¬ K . But that’s a contradition , because we cannot have both K and ¬ K . What does this mean? Well, it means that H cannot be true, because that assumption led to a contradi-tion which violated the set oF Facts 1-5 we were told about the world. (In particular it led to a contradictory statement that K ∧¬ K . ThereFore we can conclude that ¬ H . Assuming ¬ H , by (1) we get K . By (4’) we get V . By (3’) we get ¬ A . By (2’) we get ¬ R . And the only constraint leFt, (5), is satisfed because ¬ H . This leads us to a solution ( K, ¬ A, ¬ H, ¬ R,V ) which, you can check, does satisFy all the criteria 1-5. It is also a unique solution because we showed two things: ±irst, that H cannot happen. Second, that iF ¬ H then the other Four variables have uniqely assigned values, which are all a consequence oF ¬ H and Facts 1-5. ThereFore this is also the only solution. Page 1 oF 1...
View Full Document

{[ snackBarMessage ]}

0/0 PREVIEWS LEFT
Sign up to access 24/7 study resources for your classes

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern