What does $3n$ mean in set builder notation form?

Question

what does $2n$ or $3n$ mean?

e.g.:

$P= \{x : n \in \mathbb{N}, x = 3n, n< 3\}$

What does it mean?

Im still a grade 7. Dont judge me

Answer 1

Your set will contain elements of natural numbers which are less than 3, each multiplied by 3.

You could also write it $\{x : x/3 \in \mathbb N, x/3 < 3 \}$.

Answer 2

N, the set of natural numbers, is {1, 2, 3, 4, ...}. Since we also have "n< 3" n can only be 1 and 2. 3n then is 3 and 6. The set is just {3, 6}.

Answer 3

It's a multiple: $3$ times $n$.

The definition means '$P$ is a set of such $x$-es, that (for each $x$) there is such $n$ being a natural number, and less than $3$, which makes $x$ when multiplied by $3$'.

In other words: take natural numbers $n$, less then $3$, and for each such number calculate $x = 3n$; then all those results make the set $P$.

There are two such numbers: $n=1$ or $n=2$ (and possibly the third one $n=0$, depending on the definition of $\mathbb N$). That makes a set of multiples being $\{3\cdot 1, 3\cdot 2\} = \{3,6\}$ (or $\{0, 3, 6\}$, if we define $0\in\mathbb N$).
And this (one of those, precisely) becomes $P$.

Answer 4

Wikipedia can be a great resource; see their Set-builder notation article. Your exact question is addressed in the section More complex expressions on the left side of the notation.

When working with $S = {\displaystyle \{\Psi (x_{1},\ldots ,x_{n})\mid \Phi (x_{1},\ldots ,x_{n})\}}$, you can say in your head:

Define the set $S$ to be the collection of all elements of the form $\displaystyle \Psi (x_{1},\ldots ,x_{n})$
where $\Phi (x_{1},\ldots ,x_{n})$ is True.

For your example you only have one variable $n$:

$\quad \Psi (n) = 3n$

$\quad \Phi (n) : n \in \mathbb N \text{ and } n \lt 3$

$S = \{3n \, | \, n \in \mathbb N \text{ and } n \lt 3\}$, so in your mind you say,

Create the set $S$ of all objects of the form $3 \times n$ where $n$ is a natural number less than $3$.

Here you have a finite set that you can easily enumerate:

$\quad S = \{0,3,6\}$