Question

Seven is part of all of the following sets of numbers except

O irrational numbers
O integers
O rational numbers
natural numbers

Answer 1

irrational numbers

Explanation:

Answer 2

Seven isn't part of the set of irrational numbers.

Explanation:

Seven and the Set of all Natural Numbers

Start with the smallest set among the choices. The set of all natural numbers,
`\mathbb{N}`, starts with
`0` (or
`1`, for some people.) A number
`n` is in
`\mathbb{N}\!\!\!` (write
`n \in \mathbb{N}`) if and only if
`(n - 1)` is in
`\mathbb{N}\!`. Conversely, if
`n\!` is indeed in
`\mathbb{N}\!\!\!\!\!`, then
`(n + 1)` would also be in
`\mathbb{N}\!\!\!\!\!\!\!`. For
`7`:

`1 \in \mathbb{N} \implies 2 \in \mathbb{N}`.

`\vdots`.

`6 \in \mathbb{N} \implies 7 \in \mathbb{N}`.

Therefore,
`7` is indeed in the set of all natural numbers.

Seven and the Set of all Integers

Similarly, a number
`n` is in the set of integers,
`\mathbb{Z}`, if and only if either
`(n - 1)` or
`(n + 1)` is (or both are) in
`\mathbb{Z}\!\!`.

Conversely, if a number
`n` is in
`\mathbb{Z}`, then both
`(n - 1)` and
`(n + 1)` will be in
`\mathbb{Z}\!`.

It can be shown in a similar iterative way that
`7 \in \mathbb{Z}`.

Alternatively, consider the fact that the set of all natural numbers,
`\mathbb{N}`, is a subset of the set of all integers,
`\mathbb{Z}`. Therefore,
`7 \in \mathbb{N}` implies that
`7 \in \mathbb{Z}`.

Seven and the Set of all Rational Numbers

A number
`m` is a member of the set of all rational numbers
`\mathbb{Q}` if and only if there exists two integers
`p` and
`q` such that:

`\displaystyle m = (p)/(q)`.

`1` and
`7` are both integers. If
`p = 7` and
`q = 1`, then
`\displaystyle 7 = (7)/(1) = (p)/(q)`. Hence,

Alternatively, note that the set of all integers,
`\mathbb{Z}`, is a subset of the set of all rational numbers,
`\mathbb{Q}`. Therefore, the fact that
`7 \in \mathbb{Z}` would imply that
`7 \in \mathbb{Q}`.

Seven and the Set of all Irrational Numbers

A number is in the set of all irrational numbers if and only if:

• this number is in the set of all real numbers, and
• this number is not in the set of all rational numbers. (Hence "irrational.")

Therefore, the fact that
`7` is a rational number implies that it is not an irrational number.