Chapter 3.3, Problem 41ES

a.

To determine

To find: Whether the statement “ ${\forall }_x\in Z^{+}$ , ${\exists }_y\in Z^{+}$ such that $x=y+1$ ” is true or false.

b.

To determine

To find: Whether the statement “ ${\forall }_x\in Z$ , ${\exists }_y\in Z$ such that $x=y+1$ ” is true or false.

c.

To determine

To find: Whether the statement “ $\exists x\in R$ such that $\forall y\in R$ , $x=y+1$ ” is true or false.

d.

To determine

To find: Whether the statement “ ${\forall }_x\in R^{+}$ , ${\exists }_y\in R^{+}$ such that $xy=1$ ” is true or false.

e.

To determine

To find: Whether the statement “ $\forall x\in R$ , $\exists y\in R$ such that $xy=1$ “is true or false.

f.

To determine

To find: Whether the statement “ $\forall x\in Z^{+}$ and $\forall y\in Z^{+}$ , $\exists z\in Z^{+}$ such that $z=x-y$ ” is true or false.

g.

To determine

To find: Whether the statement “ $\forall x\in Z$ and $\forall y\in Z$ , $\exists z\in Z$ such that $z=x-y$ ” is true or false.

h.

To determine

To find: Whether the statement “ $\exists u\in R^{+}$ such that $\forall v\in R^{+}$ , $uv<v$ ” is true or false.