Chapter I, Problem 1E

Use these exercises to check your understanding. Answers appear at the end of the Answers to Selected Exercises.

1. Write each set in two ways: by listing its elements and by stating the property that determines membership in the set.(a) The set of integers between 6 and 12(b) The set of integers whose square is less than 17(c) The set of solutions to $x^2-81=0$
(d) The set of integer powers of 2
(e) The set of ingredients in a peanut butter and jelly sandwich

To determine

To write set in :-

(i)Roster Form

(ii)Set-Builder Form

Answer to Problem 1E

Roster Form : $\left\{7,8,9,10,11\right\}$

Set-Builder Form : { $x$ : $x$ is an integer between 6 and 12 }

Explanation of Solution

Given :

The set of integers between 6 and 12 .

Clearly, $\left\{7,8,9,10,11\right\}$ is the set of integers lying between 6 and 12 .

Hence, Roster Form : $\left\{7,8,9,10,11\right\}$

Also, { $x$ : $x$ is an integer between 6 and 12 } is the set of all integers lying between 6 and 12 .

Hence, Set-Builder Form : { $x$ : $x$ is an integer between 6 and 12 }

To determine

To write set in :-

(i)Roster Form

(ii)Set-Builder Form

Answer to Problem 1E

Roster Form : $\left\{-4,-3,-2,-1,0,1,2,3,4\right\}$

Set-Builder Form : { $x$ : $x$ is an integer whose square is less than 17 }

Explanation of Solution

Given :

The set of integers whose square is less than 17 .

  $\left\{-4,-3,-2,-1,0,1,2,3,4\right\}$ is the set of all those integers whose squares are less than 17 .

Hence, Roster Form : $\left\{-4,-3,-2,-1,0,1,2,3,4\right\}$

Also, { $x$ : $x$ is an integer whose square is less than 17 } is the set-builder form for the given problem.

To determine

To write set in :-

(i)Roster Form

(ii)Set-Builder Form

Answer to Problem 1E

Roster Form : $\left\{-9,9\right\}$

Set-Builder Form : { $x$ : $x$ is the solution of $x^2-81=0$ }

Explanation of Solution

Given :

The set of solutions to $x^2-81=0$

Let,

  $x^2-81=0$

  $\begin{array}{l}\Rightarrow x^2=81\\ \Rightarrow x=\sqrt{81}\\ \Rightarrow x=\pm 9\end{array}$ (Square root both sides)

Hence, Roster Form : $\left\{-9,9\right\}$

Also, { $x$ : $x$ is the solution of $x^2-81=0$ } is the set-builder form .

To determine

To write set in :-

(i)Roster Form

(ii)Set-Builder Form

Answer to Problem 1E

Roster Form : $\left\{\mathrm{....},\frac{1}{4},\frac{1}{2},1,2,4,\mathrm{....}\right\}$

Set-Builder Form : { $x$ : $x$ is an integral power of 2 }

Explanation of Solution

Given :

The set of integer powers of 2 .

First of all, the set of integers is $Z=\left\{\mathrm{....},-2,-1,0,1,2,\mathrm{....}\right\}$

Hence, the set of integral powers of 2 will be $\left\{\mathrm{....},2^{-2},2^{-1},2^0,2^1,2^2,\mathrm{....}\right\}$

Hence, Roster Form : $\left\{\mathrm{....},2^{-2},2^{-1},2^0,2^1,2^2,\mathrm{....}\right\}$ = $\left\{\mathrm{....},\frac{1}{4},\frac{1}{2},1,2,4,\mathrm{....}\right\}$

Also, Set-Builder Form : { $x$ : $x$ is an integral power of 2 }

To determine

To write set in :-

(i)Roster Form

(ii)Set-Builder Form

Answer to Problem 1E

Roster Form : { Bread, Peanut Butter, Jelly }

Set-Builder Form : { $x$ : $x$ is an ingredient in peanut butter and jelly sandwich }

Explanation of Solution

Given :

The set of ingredients in a peanut butter and jelly sandwich .

This delicious sandwich is made up of bread, peanut butter and jelly .

Hence, Roster Form : { Bread, Peanut Butter, Jelly }

Also, Set-Builder Form : { $x$ : $x$ is an ingredient in peanut butter and jelly sandwich }