CMSC 250 Fall 2001 Homework 4 solutions

The truth set for the situations will be:
1. {2,4,6,8}
2. $\{x \in {\bf {R}}\vert x \leq -5\}$
3. $\{ ta,te,ti,to,tu\}$ OR $\{ abc\,\, \vert\,\, a = 't', b \in \{'a','e','i','o','u'\}, \\ c =\{\hbox{strings of 0 or more letters}\}\}$
4. $\{1,2,3,4,6,12\}$
1. $\exists x \in {\bf {Z}}$ , such that $\forall y \in {\bf {Z}}, y \neq \sqrt{x}$
2. $\exists x \in {\bf {R}}$ , such that $x^2 \in{\bf {Z}}$ .
3. $\forall$ bases b, b belongs to us. OR $\forall b\in \{bases\} B(b)$ , where = ``b belongs to us''.
4. $\forall$ dogs d , d has his day. OR $\forall d\in \{dogs\}, \exists y\in \{days\} D(d,y)$ , where = ``d has his y''.
5. $\forall r \in {\bf {Q}}, r \in {\bf {R}}$
Negations of the given statements using informal language
1. There was a movie that came out this summer, which was well written.
2. No student is taking both CMSC 250 and CMSC 251 this semester.
3. Someone remembered to turn the lights off last night.
4. Someone has been able to fix the parking problem on campus.
5. There exists an integer whose square is not an integer.
6. All continuous functions have derivatives at every point.
1. $\forall x \in {\bf {R}}, \exists y \in {\bf {R}}$ such that $y = \sqrt[3]{x}$
2. $\exists y \in {\bf {R}}$ , such that $\forall x \in {\bf {Z}}^{nonneg}, y^x=y$
3. $\forall x,y \in {\bf {R}}, \exists z \in {\bf {R}}$ , such that $\mid x-z \mid = \mid y-z \mid$
4. $\exists$ a person x such that $\forall$ people y, x gets along with y. OR $\exists p\in \{people\}, \forall y\in\{people\},\,\, G(x,y)$ , where ``x gets along with y''.
The negation of the given statements using formal language:
1. $\forall x \in {\bf {R}}^{+}, \exists y \in {\bf {R}}^{+}, y = x^2$
2. $\exists x \in {\bf {Z}}$ , such that $x+x=x \ast x$
3. $\forall x \in {\bf {Z}}\forall k\in{\bf {Z}}, x^2 \neq 4k+3$ .
1. Contrapositive: $\forall x \in {\bf {Z}}$ , If x is not even, then is not even.
  Converse: $\forall x \in {\bf {Z}}$ , if x is even, then is even.
  Inverse: $\forall x \in {\bf {Z}}$ ,If is not even, then x is not even.
2. Contrapositive: $\forall x \in {\bf {Z}}$ ,If x does not have 3 factors, then $\sqrt{x}$ is not a prime
  Converse: $\forall x \in {\bf {Z}}$ , If x has 3 factors, then $\sqrt{x}$ is a prime.
  Inverse: $\forall x \in {\bf {Z}}$ , If $\sqrt{x}$ is not a prime, then x does not have 3 factors.
3. Contrapositive: $\forall \epsilon > 0,\exists N_0 > 0$ such that If $\mid a_n$ -L $\mid \ge \epsilon$ then n $\le N_0$ .
  Converse: $\forall \epsilon > 0,\exists N_0 > 0$ such that If $\mid a_n-L \mid < \epsilon$ then n .
  Inverse: $\forall \epsilon > 0,\exists N_0 > 0$ such that If n $\le N_0$ , then $\mid a_n-L\mid \ge \epsilon$ .