Name (PRINTED):

Student ID #:

Section # (or TA's:
name and time)  

CMSC 250 Exam #1 Thursday, Oct. 11, 2001

Write all answers legibly in the space provided. The number of points possible for each question is indicated in square brackets - the total number of points on the exam is 150, and you will have exactly 1.5 hours to complete this exam. You may not use calculators, textbooks or any other aids during this exam.
  1. [20 pnts.] Determine if the following pair of statements is logically equivalent. Use 1 for ``true'' and 0 for ``false'' to create the complete truth table. If they are not equivalent, indicate which lines indicate that they are not equivalent; if they are equivalent, mark the columns which show that.


    (Yes or No) These statements are logically equivalent.
  2. [10 pnts.] Fill in all of the blanks to show the subtraction of numbers (through binary addition using 8-bit 2's-compliment).

    Solve the problem of 9 - 5 = ??.
    The 8-bit binary representation of the  
    decimal value 9 =  
    The 8-bit 2's complement representation of  
    the decimal value -5 =  
    Adding the previous lines you get an  
    8-bit 2's complement =  
    The decimal representation of the sum on the  
    on the previous line =  
  3. [20 pnts.] Use the handout of the ``Logical Equivalence Rules'' and the ``Rules of Inference'' to prove the following. It is a Valid Argument - you only need to prove that it is.
    P1 $p \vee q $
    P2 $ \sim (\sim r \wedge q) $
    P3 $ \sim p \vee r $
      $ --------------$
      therefore $r$
    line Statement Reason Line #s
    1      
           
    2      
           
    3      
           
    4      
           
    5      
           
    6      
           
    7      
           
    8      
           
    9      
           
    10      
           
    11      
           
    12      
           
    13      
           
    14      
           
  4. [20 pnts.] For each of the following English sentences, translate the meaning into formal notation using the logic symbols ($\exists$, $\forall$, $\wedge $, $\vee $, $\sim $, and $\rightarrow$). In addition to these, you may also use mathematical, grouping and set notations symbols as needed. On the next line write the negation of the original statement using formal notation.
    There is at least one dog who lives in some house.
    Domains: D = {all dogs}, H = {all houses}
    Predicate: L(x,y) = ``dog x lives in house y''
    statement:
     
    negation:
     
     
    For every integer there is an integer that is less than it.
    Domain: Z = {all integers}
    Predicate: L(x,y) = ``integer x is less than integer y''
    statement:
     
    negation:
     
     
    Some of the real numbers can be written as a quotient of two integers.
    Domains: R = {all reals}, Z = {all integers}
    Predicate: Q(r,a,b) = ``real r is the quotient of integers a and b''
    statement:
     
    negation:
     
     
    There is at most one cat that lives inside (in a house). [means: All other cats live outside (not in a house).]
    Domains: C = {all cats}, H = {all houses}
    Predicates: L(x,y) = ``cat x lives in house y''
    statement:
     
    negation:
     
     
  5. [10 pnts.] Use an Euler diagram to determine if each of the following represents a valid argument. Make sure to label the parts of the diagram.
    All dogs have fur.
    All dogs have fleas.
    -----
    therefore: All things with fur have fleas.
     
    Circle One: Valid Invalid
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
    Only birds can fly.
    No birds are mammals.
    -----
    therefore: No mamals can fly.
     
    Circle One: Valid Invalid
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
  6. [20 pnts.] Use the handout of the ``Logical Equivalence Rules'' and the ``Rules of Inference'' to prove the following. It is a Valid Argument - you only need to prove that it is. You may also assume the Domain is not empty (it has at least one member).
    P1 $\exists x \in D, M(x) \wedge Q(x)$
    P2 $\forall y \in D, M(y) \rightarrow R(y)$
    P3 $\forall z \in D, \sim (\sim P(z) \wedge Q(z))$
      $ --------------$
      therefore $\exists w \in D, R(w) \wedge P(w)$
    line Statement Reason Line #s
    1      
           
    2      
           
    3      
           
    4      
           
    5      
           
    6      
           
    7      
           
    8      
           
    9      
           
    10      
           
    11      
           
    12      
           
    13      
           
    14      
           
  7. [10 pnts.]Give an example of a predicate $P(x,y,z)$ that makes the statement

    \begin{displaymath}\forall x\in D, \forall y\in D, \exists z\in D,\,\, P(x,y,z)\end{displaymath}

    true if $D$ is the set of rational numbers, and false if D is the set of integers.












  8. [20 pnts.] Use the unique prime factorization theorem to prove: $\sqrt[3]{7} \not \in Q$.
  9. [20 pnts.]Prove or give a counter-example to disprove the following claim: If $a$, $b$, $c$, $m$, and $n$ are positive integers and that $a \equiv b$ mod $n$, $b \equiv c$ mod $m$ and $m\vert n$, then $m \vert(a-c)$.

About this document ...

This document was generated using the LaTeX2HTML translator Version 99.1 release (March 30, 1999)

Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.

The command line arguments were:
latex2html -show_section_numbers -split 0 -no_navigation -no_footnode ex1fall01.tex

The translation was initiated by John Arras on 2002-03-08

John Arras
2002-03-08

Web Accessibility