# What if there are multiple conditions in set-builder notation?

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...

1. all integers greater than 0 but less than 6
2. all positive even integers less than 11
3. 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.

• What do you mean with "interval of values"? The set [1, 0] has infinite elements. – Pel de Pinda Jun 29 '17 at 11:36
• @PeldePinda The question has been updated. – Soha Farhin Pine Jun 29 '17 at 12:37
• You use logic to make several conditions. $\land$,$\lor$,$\top$,$\bot,=$ for example, and stick predicates between them (if you use binary operations infix) and parentheses to avoid ambiguity. You use $\in$ to say which set the variable belongs to. And you can bind the variable using stuff like $\forall,\exists$. – Emil Jun 29 '17 at 13:47

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

$$\{k^2| k\in\mathbb N\land k^2<101\}$$

• The question has been updated. – Soha Farhin Pine Jun 29 '17 at 12:37
• @SohaFarhinPine So was the answer. – 5xum Jun 29 '17 at 12:41
• Thanks! And 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. – Soha Farhin Pine Jul 1 '17 at 10:50
• I just wanted to know how set-builder notation works in general. I don't know exactly why you used that triangl-ish sign to separate conditions. I don't why other signs were used. Are there any other separators? I didn't understand why you used that inverted 'E' before 'k' in the #1 perfectsquares example and why this time you used colon as the separator. What are these signs and symbols? What do they actually do? – Soha Farhin Pine Jul 1 '17 at 15:49
• I'm a 8th grader, and I've never needed to learn logic. The general guidelines aren't a book worth of text. I already know what set-builder notation is; I just want to learn high-level stuff. Your comment was really short, and yet it helped me understand a lot. So no---all this explanation can really be wrapped up into a single answer. Answers can be really long as well (and many good answers are). – Soha Farhin Pine Jul 2 '17 at 0:02