TITLE: What if there are multiple conditions in set-builder notation?
QUESTION [6 upvotes]: I've recently started learning about sets, and most examples of set-builder notation I've encountered:

are infinite
$A={\text{{set of all positive numbers}}}$
need only one condition (to define all the elements of the set)
$A={\text{\{}x:x>0}\text{\}}$  
have $x$ (the number to represent each element in the set) on LHS and a condition (to define all $x$) on the RHS with a colon in between

The first bullet, I suppose, explains the other two bullets. The simple examples of set-builder notations I saw were all infinite. That's the main point.

What if a set is not infinite, but rather can be represented as a
  interval of values? How to show such a set in the set-builder notation?

Then the set will have multiple conditions and the format of the notation will have to change.
For example, 
$$A={\text{{all numbers greater than zero}}} \\ B={\text{{all integers between 1 and 20}}}$$
A infinite set is like a "super" universal set which can have subsets. Set $A$ is a universe. A universe of all numbers greater than zero i.e. all positive integers. Its subset $B$ contains the first 20 positive integers i.e. $[1, 20\text{]}$. It's a small piece of a gigantic universe: a closed interval with endpoints.

How will you show a set of...

all integers greater than 0 but less than 6
all positive even integers less than 11
all perfect squares less than 101

in set-builder notation?


EDIT: @5xum There were two questions. You missed the broad one. And it will be really helpful if you would have had explained how this format works. Otherwise, your answer seems like a bunch of back-of-book school textbook maths solutions. General guidelines help much more than complete solutions to specific problems.

REPLY [7 votes]: The set $\{x| x\in \mathbb N\land x<1000\}$ is finite and it can easily be presented in the set builder notation...

All integers greater than $0$ but less than $6$:
$$\{n| n\in\mathbb N\land n>0 \land n<6\}$$

All positive even integers less than $11$:
$$\{n| n\in\mathbb N\land n<11\land (\exists k\in\mathbb N: n=2k)\}$$
or, alternatively, 
$$\{2k| k\in\mathbb N\land 2k<11\}$$

All perfect squares less than $101$:
$$\{n| n\in\mathbb N\land n<101\land (\exists k\in\mathbb N: n=k^2)\}$$
or, again, alternatively as
$$\{k^2| k\in\mathbb N\land k^2<101\}$$