{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Assignment1-solutions - MACM 101 Discrete Mathematics I...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MACM 101 — Discrete Mathematics I Exercises on Propositional Logic. Due: Tuesday, Septem- ber 29th (at the beginning of the class) SOLUTIONS 1. Construct a truth table for the following compound proposition: ( p q ) ( p q ) Solution p q ( p q ) ( p q ) 0 0 0 0 1 1 1 0 1 1 1 1 2. Which of the following statements are tautologies? ( p q ) ( ¬ p q ) ( p q ) ( ¬ ( p ∧ ¬ q )) ( p q ) ( p q ) Solution Method 1: Use truth tables. Method 2: Reason about it: 1) Yes, each of ( p q ) and ( ¬ p q ) is false if and only if p is true and q is false. 2) Yes, it follows from 1), De Morgan’s law and double negation law: ( p q ) ( ¬ p q ) ( p q ) ( ¬¬ ( ¬ p q )) ( p q ) ( ¬ ( ¬¬ p ∧ ¬ q )) ( p q ) ( ¬ ( p ∧ ¬ q )) 3) No, if we set p = 0 , q = 1 then the formula turns into 0. 3. Show that ( p r ) ( q r ) and ( p q ) r are logically equivalent. Method 1: Use truth table. 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Method 2: Use the laws of logic. ( p r ) ( q r ) (definition of ) ( ¬ p r ) ( ¬ q r ) (distributivity) ( ¬ p ∧ ¬ q ) r (De Morgan) ( ¬ ( p q )) r (definition of ) ( p q ) r 4. Show that ( a b c d e ) f and ( a b c d e ) f are not logically equivalent Solution If we set a = b = c = d = 0 , e = 1 , f = 0 then we have the first statement to be 1 and the second to be 0.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

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