CMSC 250 Homework 4 Fall 2001

Due Wed Sept 26 at the beginning of your discussion section.

1. Write the truth set for each of the following situations, where the domain is specified by D, and P(x) specifies the predicate.
   1. D is the set of all even integers. P(x) = ``x is more than 1 and less than 9''.

   2. D is the set of all real numbers. P(x) = ``x is no greater than -5''.

   3. D is the set of all strings of 1 or more lowercase letters. P(x) = ``the first letter of x is 't' and the second letter is a vowel''.

   4. D is the set of all positive integers. P(x) = ``x is a factor of 12''.

2. Translate the following informal statements to formal language.
   1. There is a positive integer that has a square root that is not an integer.

   2. Some squares of real numbers are integers.

   3. All of your bases belong to us.

   4. Every dog has his day.

   5. Every rational number is a real number.

3. Write the negations of the following statements using informal language.
   1. Every movie that came out this summer was poorly written.

   2. There is a student taking CMSC 250 and CMSC 251 this semester.

   3. No one remembered to turn the lights off last night.

   4. No one has been able to fix the parking problem on campus.

   5. The square of every integer is an integer.

   6. There are some continuous functions that do not have derivatives at every point.

4. Write the following statements containing multiple quantifiers using formal language.
   1. For every real number x, there is a real number y such that y is the cube root of x.

   2. There is a real number such that every positive integer power of itself is itself.

   3. Given any two points on the real line, there is a point that is equidistant between them.

   4. Somebody can get along with anybody.

5. Write the negations of the following statements using formal language.
   1. $\exists x\in{\bf {R}}$, with $x > 0$, such that $\forall y\in{\bf {R}}$, with $y > 0$, $y \neq x^{2}$.

   2. $\forall x\in{\bf {Z}}$, $x + x \neq x * x$.

   3. $\exists x\in{\bf {Z}}$, such that $x^{2} = 4k+3$ for some $k\in{\bf {Z}}$.

6. Write the contrapositive, the inverse, and the converse of each of the following statements:
   1. $\forall x\in{\bf {Z}}$, if $x^{2}$ is even, then x is even.

   2. $\forall$ positive $x\in{\bf {Z}}$, if $\sqrt{x}$ is a prime then x has 3 factors.

   3. $\forall\epsilon>0$, $\exists N_{0} > 0$ such that if $n > N_{0}$, then $\vert a_{n} - L\vert <\epsilon$.

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