Chapter 7.1, Problem 1E

(a)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

Upper bound exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{x\in ℝ:x^2<10\}$ .

The set that is given is $\{x\in ℝ:x^2<10\}$ so we can see that $x<\sqrt{10}<10$ therefore the two upper bounds are $\sqrt{10}$ and $10$ .

Hence, upper bounds exist for the given set.

(b)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

Upper bounds exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{\frac{1}{3^x}:x\in \mathbb{N}\}$ .

The set that is given $\{\frac{1}{3^x}:x\in \mathbb{N}\}$ so we can see that the sequence is decreasing so the upper bounds are 1/3 and 1.

Hence, upper bounds exist for the given set.

(c)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

Upper bounds exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{x\in ℝ:x+\frac{1}{x}<5\}$ .

The set that is given is $\{x\in ℝ:x+\frac{1}{x}<5\}$ so we can see that $x<5$ so surely 5 is one upper bounded value.

Again, if we solve $x+\frac{1}{x}=5$ then the result will be $x=\frac{1}{2}(\sqrt{21}+5)$ which is the other upper bound value.

Hence, upper bounds exist for the given set.

(d)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

Upper bounds exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{x\in ℝ:x^2+2x-3<0\}$ .

The set that is given is $\{x\in ℝ:x^2+2x-3<0\}$ so we can see that $x^2+2x<3$ so surely 3 is one upper bounded value.

Again, if we solve $x^2+2x-3=0$ then the result will be $x=1$ which is the other upper bound value.

Hence, upper bounds exist for the given set.

(e)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

No upper bounds exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{x\in ℝ:x^2-3>0\}$ .

The set that is given is $\{x\in ℝ:x^2-3>0\}$ so we can see that $x^2>3$ which says that $x$ can be any number greater than $\sqrt{3}$ .

Hence, no upper bounds exist for the given set.

(f)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

Upper bounds exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{x\in ℝ:x^2<0$ and $x^2-3>0\}$ .

The set that is given is $\{x\in ℝ:x^2<0$ and $x^2-3>0\}$ so by definition of sets 0 will be the upper bound.

Again, by solving $x^2-3<0$ we get $-\sqrt{3}$ which is the another upper bound.

Hence, upper bounds exist for the given set.

(g)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

No upper bounds exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{2^{-x}x\in ℝ\}$ .

The set that is given is $\{2^{-x}x\in ℝ\}$ so by definition for negative $x$ , $2^{-x}$ is positive and its greater than any real number.

Hence, no upper bounds exist for the given set.

(h)

To determine

To find: That if there exist any upper bounds for the following set.

Answer to Problem 1E

Upper bounds exist for the given set.

Explanation of Solution

Given Information:

The given set is $\{x\in ℝ:x-10<\log x\}$ .

The set that is given is $\{x\in ℝ:x-10<\log x\}$ if we solve numerically $x-10=\log x$ we get $x$ to be approximately equal to 0.000045402, 12.528 therefore $x-10<\log x$ for the value 0.000045402< $x<12.528$ . The existing upper bounds will be 12.6 and so on.

Hence, upper bounds exist for the given set.