Write formal descriptions of the following sets.

1. The set containing the numbers$\mathbf{\text{1,10}}$, and$\mathbf{\text{100}}$
2. The set containing all integers that are greater than$\mathbf{\text{5}}$
3. The set containing all natural numbers that are less than$\mathbf{\text{5}}$
4. The set containing the string aba
5. The set containing the empty string
6. The set containing nothing at all

A set is amathematical modelfor a collection of different things which containselementsor members which are numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other set.

a.

The three numbers in this set denoting this set as$\{1,10,100\}$.It meansncontains$10$and$10$to the power of$\mathrm{m}$contain the series$0$ , $1$,and$2$is as shown below.

Therefore, the given description is defined in terms of set as follows:

$\{\mathrm{n}:\mathrm{n}={10}^{\mathrm{m}}\mathrm{for}\mathrm{some}\mathrm{n}\in \{0,1,2\}\}$

b.

It isgiven that aset contains all the integers greater than $5$.

Thus,ncontains only integers greater than5is as shown below:

$\{\mathrm{n}:\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{integer}\mathrm{and}\mathrm{n}>5\}$

c.

Given that theset contains all the integers less than5.

Thus, thencontains only those integers which are less than5as shown below:

_{$\{\mathrm{n}:\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{natural}\mathrm{number}\mathrm{and}\mathrm{n}<5\}$}

d.

Given that theset contains only a single element and that is the string ‘aba’

Thus, n contains only the string ‘aba’ is shown below:$\{\mathrm{n}:\mathrm{n}\mathrm{contains}\left\{\mathrm{aba}\right\}\}$

e.

According to the given statement, the set contains the empty string. Empty is denoted by$\mathrm{\epsilon }$

Thus, the set which contains only the empty string is as shown below:$\left\{\mathrm{\epsilon }\right\}$

f.

According to the given information, the given set contains nothing, which means the set contains only null. Null is denoted by$\mathrm{\varphi }$

Hence, the set containing null is represented as $\mathrm{\varphi }$.