Practice Exam1 - 347-G1 Fall 2009 Practice Exam 1 Solutions...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
347-G1, Fall 2009 Practice Exam 1 Solutions 1 1. (15 points) For each of the claims below, rewrite them in logical notation, negate them and then write out an English sentence which expresses the negated claim (without using words of negation). (a) For every natural number n we can find an x R so that x < 1 n 3 . Solution. In logical notation: ( n N )( x R ) x < 1 n 3 ¬ ( n N )( x R ) x < 1 n 3 ( n N ) ¬ ( x R ) x < 1 n 3 ( n N )( x R ) x 1 n 3 The negation is: There exists a natural number n so that for all real numbers x , x 1 n 3 . (b) There exists M R so that for all real numbers x, y M , if x < y then f ( x ) < f ( y ) . Solution. In logical notation: ( M R )( x, y M ) x < y f ( x ) < f ( y ) ¬ [( M R )( x, y M ) x < y f ( x ) < f ( y )] ( M R ) ¬ [( x, y M ) x < y f ( x ) < f ( y )] ( M R )( x, y M ) x < y f ( x ) f ( y ) The negation is: For all real numbers M , there are x, y M so that x < y and f ( x ) f ( y ). (c) For all real numbers x , if x 2 < 2 x holds, then we must have x > 2 or x < 0 .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
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