Chapter 2.CT, Problem 1CT

Chapter Test

Use an alternative method to express each set.

a. $\left\{101,102,103,104,\mathrm{.....}\right\}$

b.{x:x is a month in the year}

c.{y:y is a person in your math class and also more than 100 years old}

To determine

a)

To express:

An alternative method of each set

$\left\{101,102,103,104,\mathrm{.....}\right\}$

Answer to Problem 1CT

Solution:

$S=\left\{n:n\text{ is a natural number greater than 100}\right\}$

Explanation of Solution

Definition of sets and elements:

A collection of objects is called a set and the individual objects in this collection are called elements or members of the set. We can use uppercase letter to denote the set and the lowercase letter to denote the elements.

For example:

$S=\left\{a,b,c,d\right\}$

Where S is the set and a, b, c, d are the elements

Sets can be represented by using two methods. They are

i) listing method

ii) set-builder notation

Listing method:

The elements in the set can be written as a list where the elements are separated by the commas and enclosed within the brackets.

Set-builder notation:

A shorthand used to write sets, if all the elements of set have some common characteristics. It is also used to define a set with infinite number of elements.

For example:

We can represent a natural numbers using set-builder notation,

$N=\left\{x:xisanaturalnumber\right\}$

We can read the above expression as “$N$ is the set of all $x$ such that $x$ is a natural number”.

Given:

$\left\{101,102,103,104,\mathrm{.....}\right\}$

The given set is expressed in listing method.

So we can express that equation in the set-builder notation which is the alternative method.

$S=\left\{n:n\text{ is a natural number greater than }100\right\}$

Here “$S$ is the set of all $n$ such that $n$ is a natural number greater than $100$”.

To determine

b)

To express:

An alternative method of each set

$\left\{\text{x:x is a month in the year}\right\}$

Answer to Problem 1CT

Solution:

$\text{M={January, February, March,}\mathrm{...}\text{, December}}$

Explanation of Solution

Definition of sets and elements:

A collection of objects is called a set and the individual objects in this collection are called elements or members of the set. We can use uppercase letter to denote the set and the lowercase letter to denote the elements.

For example:

$S=\left\{a,b,c,d\right\}$

Where S is the set and a, b, c, d are the elements

Sets can be represented by using two methods. They are

i) listing method

ii) set-builder notation

Listing method:

The elements in the set can be written as a list where the elements are separated by the commas and enclosed within the brackets.

Set-builder notation:

A shorthand used to write sets, if all the elements of set have some common characteristics. It is also used to define a set with infinite number of elements.

For example:

We can represent a natural numbers using set-builder notation,

$N=\left\{\text{x:x is a natural number}\right\}$

We can read the above expression as “$N$ is the set of all $x$ such that $x$ is a natural number”.

Given:

$\left\{\text{x:x is a month in the year}\right\}$

The given set is expressed in set-builder notation method.

So we can express that equation in the listing which is the alternative method.

$M\text{={January, February, March,}\mathrm{...}\text{, December}\}$

To determine

c)

To express:

An alternative method of each set

$\left\{\text{y:y is a person in your math class and also more than 100 years old}\right\}$

Answer to Problem 1CT

Solution:

$C=\left\{\right\}or\varphi$

Explanation of Solution

Definition of sets and elements:

A collection of objects is called a set and the individual objects in this collection are called elements or members of the set. We can use uppercase letter to denote the set and the lowercase letter to denote the elements.

For example:

$S=\left\{a,b,c,d\right\}$

Where S is the set and a, b, c, d are the elements

Sets can be represented by using two methods. They are

i) listing method

ii) set-builder notation

Listing method:

The elements in the set can be written as a list where the elements are separated by the commas and enclosed within the brackets.

Set-builder notation:

A shorthand used to write sets, if all the elements of set have some common characteristics. It is also used to define a set with infinite number of elements.

For example:

We can represent a natural numbers using set-builder notation,

$N=\left\{\text{x:x is a natural number}\right\}$

We can read the above expression as “$N$ is the set of all $x$ such that $x$ is a natural number”.

Given:

$\left\{\text{y:y is a person in your math class and also more than 100 years old}\right\}$

The given set is expressed in set-builder notation method.

So we can express that equation in the listing which is the alternative method.

$C=\left\{\right\}or\varphi$