If

$$K_1\ and\ K_2$$

are disjoint nonempty compact sets, show that there exist

$$k _ { i } \in K _ { i }$$

such that

$$0 < \left| k _ { 1 } - k _ { 2 } \right| = \inf \left\{ \left| x _ { 1 } - x _ { 2 } \right| : x _ { i} \in K _ { i } \right\}.$$

Graph the inequality.

$$3 x \geq - y + 3$$

The length of a rectangle is 3 in. more than its width. The perimeter of the rectangle is 30 in. Define a variable for the width.

Prove, for every two disjoint compact sets S and T of real numbers, that there exist disjoint open sets $U$ and $V$ such that $S \subseteq U$ and $T \subseteq V$.

Prove, for every compact set S of real numbers and every real number $r \notin S$, that there exist disjoint open sets $U$ and $V$ such that $S \subseteq U$ and $r \in V$.

Find the product. -3(5)

Let A and B be disjoint compact subspaces of the Hausdorff space X. Show that there exist disjoint open sets U and V containing A and B, respectively.

a) Prove that if A and B are two nonempty disjoint open sets of real numbers, then $A \cup B \neq \mathbf{R}$.

b) Prove that the only sets of real numbers that are both open and closed are $\mathbf{R}$ and $\emptyset$.

A particular 12 V car battery can send a total charge of 84 A$\cdot$h (ampere-hours) through a circuit, from one terminal to the other. How many coulombs of charge does this represent?

Graph the function. h(x) = -1

Let S contain four sample points,

$$E _ { 1 } , E _ { 2 } , E _ { 3 } , \text { and } E _ { 4 }$$

. Let A and B be the events

$$\left\{ E _ { 1 } , E _ { 2 } , E _ { 3 } \right\} \text { and } \left\{ E _ { 2 } , E _ { 4 } \right\}$$

, respectively. Give the sample points in the following events:

$$A \cup B , A \cap B , \overline { A } \cap \overline { B } , \text { and } \overline { A } \cup B$$

.

Use a determinant to find the area of the figure with the given vertices. (−3, 5), (2, 6), (3, −5)

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.

$$\left\{ \begin{array} { c } { ( y - 3 ) ^ { 2 } = x - 2 } \\ { x + y = 5 } \end{array} \right.$$

Find dy/dx.

$$y = e ^ { \left( x - e ^ { 3 x } \right) }$$