91Ó°ÊÓ

1\. No birds except ostriches are at least 9 feet tall. 2\. There are no birds in this aviary that belong to anyone but me. 3\. No ostrich lives on mince pies. 4\. I have no birds less than 9 feet high.

In 32-38, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Any even integer equals twice some integer.

Give an example to show that a universal conditional statement is not logically equivalent to its inverse.

Give an example to show that a universal conditional statement is not logically equivalent to its inverse. Earning a grade of \(\mathrm{C}\) - in this course is a sufficient condition for it to count toward graduation.

Give an example to show that a universal conditional statement is not logically equivalent to its inverse. Being divisible by 8 is a sufficient condition for being divisible by \(4 .\)

Indicate which of the following statements are true and which are false. Justify your answers as best you can. a. \(\forall x \in \mathbf{Z}^{+}, \exists y \in \mathbf{Z}^{+}\)such that \(x=y+1\). b. \(\forall x \in \mathbf{Z}, \exists y \in \mathbf{Z}\) such that \(x=y+1\). c. \(\exists x \in \mathbf{R}\) such that \(\forall y \in \mathbf{R}, x=y+1\). d. \(\forall x \in \mathbf{R}^{+}, \exists y \in \mathbf{R}^{+}\)such that \(x y=1\). e. \(\forall x \in \mathbf{R}, \exists y \in \mathbf{R}\) such that \(x y=1\). f. \(\forall x \in \mathbf{Z}^{+}\)and \(\forall y \in \mathbf{Z}^{+}, \exists z \in \mathbf{Z}^{+}\)such that \(z=x-y\). g. \(\forall x \in \mathbf{Z}\) and \(\forall y \in \mathbf{Z}, \exists z \in \mathbf{Z}\) such that \(z=x-y\). h. \(\exists u \in \mathbf{R}^{+}\)such that \(\forall v \in \mathbf{R}^{+}, u v

Let \(P(x)\) and \(Q(x)\) be predicates and suppose \(D\) is the domain of \(x\). In 53-56, for the statement forms in each pair, determine whether (a) they have the same truth value for every choice of \(P(x), Q(x)\), and \(D\), or (b) there is a choice of \(P(x), Q(x)\), and \(D\) for which they have opposite truth values. $$ \begin{aligned} &\forall x \in D,(P(x) \wedge Q(x)), \text { and } \\ &(\forall x \in D, P(x)) \wedge(\forall x \in D, Q(x)) \end{aligned} $$