Question

Can someone help me understand builder notation in Algebra? I can't understand the concept at all...

Answer 1

set-builder notation?

Answer 2

Yes

Answer 3

Okay, let's have a look at a rather easy example...
This set:
\[\Large \{2,4,6,8,10...\}\]
This is the roster form of the set of positive even numbers, right?

Answer 4

Well,I think u must watch this

Answer 5

https://www.youtube.com/watch?v=xnfUZ-NTsCE

Answer 6

Right*

Answer 7

@LifeIsADangerousGame

https://www.youtube.com/watch?v=xnfUZ-NTsCE

Answer 8

kk, I'll watch that video Udit

Answer 9

Nice video.

Answer 10

Well, yes

Answer 11

Okay, it makes more sense after watching the video, Is x the numbers between the brackets?

Answer 12

Yes... the numbers between the curly brackets should be all the values x could possibly take.
And that set we have above was the second set the guy handled... he defined it as
\[\Large \{2x \ \left| \ x\in \mathbb{N} \right.\}\]

Answer 13

Well do u know khanacademy

Answer 14

Personally, I prefer
\[\Large \{x \ \left| \ x = 2k \ ; \ k\in \mathbb{N} \right.\}\]

Answer 15

So the numbers is the brackets is what x could be, not what x is going to be if you take one of the numbers from inside of the brackets and plug it into x?

Answer 16

Think of it this way....
\[\Large \{x \ \left| \ \color{red}{\text{an appropriate description so that no value is missed}} \right. \}\]

Answer 17

Here's a trickier one: It's related to his last example:
\[\Large \{2,3,5,6,7,8,10,11, 12...\}\]

Answer 18

Basically, we're listing all the positive integers (natural numbers) which are NOT perfect squares... how do we proceed in writing this in set-builder?

Answer 19

So... let's have a little step-by-step;
\[\Large \color{red}{\{x \ |}\]
This means "the set of all x such that..."

Answer 20

Okay, so now, we describe ALL the elements of that set... how would you describe them?

Answer 21

its plus 1 for a few numbers then plus 2 and so on?

Answer 22

Well, not like that... something that's more general... they're all positive integers (natural numbers) aren't they?

Answer 23

They're all natural numbers and they go on until infinity?

Answer 24

"They're all natural numbers" is enough... let's put that down...
\[\Large \color{red}{\{x \ \left| \ \color{green}{x\in \mathbb{N} }\right. } \]

Answer 25

Are we done here?

Answer 26

I don't think so, its all natural numbers, but there's a patten I think

Answer 27

The answer is no...we can't end it here quite yet... if we do, the set would erroneously include ALL natural numbers, including 1, 4, 9, 16 etc... which aren't supposed to be in the set...

Answer 28

Just a reminder: this:
\[\Large \{2,3,5,6,7,8,10,11, 12...\}\]
is our set... set of all natural numbers which aren't perfect squares...

Answer 29

Maths is like a language itself... how do you say "x is not a perfect square" in mathematical-symbols?

Answer 30

x^1?

Answer 31

Much like a language, there are 'sentences' and 'phrases' in maths... and that's just a phrase :D

"x is not a perfect square" is a sentence... it must contain an equation or an inequality to be a sentence :D

Answer 32

This is a phrase:
\[x + y\] (sum of x and y)

And this is a sentence:
\[x +y= z\](The sum of x and y is z.)

Answer 33

Okay, to say that x is not a perfect square, we can write...
\[\Large \color{red}{\{x \ \left| \ \color{green}{x\in \mathbb{N} \ ; \color{blue}{ \ x \ne k^2} }\right. }\]

Is that enough?

Answer 34

No we need to do 2 < x < oo (infinity) I think?

Answer 35

The fact that we have \(\large x \in \mathbb{N}\) already says that x goes to infinity... it's important not to be redundant...

Answer 36

Oh then maybe just 2 < x?

Answer 37

Not quite... there's only one more thing to add... keep in mind that we just added a new variable, k.
We now have to define what k is.

In particular, x is not a perfect square if x is not equal to k^2 for any integer k, right?

Answer 38

So finally, we add this detail:
\[\Large \color{red}{\{x \ \left| \ \color{green}{x\in \mathbb{N} \ ; \color{blue}{ \ x \ne k^2} } \ ; \ k \in \mathbb{N}\right. } \ \}\]
And we're done...

Answer 39

So we read this as:
The set of all x
such that x is a natural number
where x is not equal to k^2
for any natural number k.

Indeed, that gives us all the positive integers which are not perfect squares.

Answer 40

By the way, there is more than one way to say that... there is something far simpler :D
But (in my opinion) trickier to spot... one can also have:
\[\Large \{x \ \left| \ \color{blue}{\sqrt x \notin \mathbb{N}}\right.\}\]

Answer 41

Or the set of all x where the square root of x is NOT an integer.

Answer 42

Wait, sorry...
\[\Large \{x \ \left| \ \color{green}{x\in \mathbb{N} } \ \color{blue}{ \ ; \sqrt x \notin \mathbb{N}}\right.\}\]

Answer 43

^ That was simpler...

For most sets, there is more than one way to write it in set-builder notation... what's important is that they all refer to one set.

Answer 44

Don't worry if that confused you... that was a confusing set to begin with...
Can you try this:
\[\Large \{1,3,5,7,9...\}\]?
Clearly the set of all positive odd numbers... how would you write this in set-builder?

Answer 45

I apologize for not answering, I felt sick and had to brb. I'll try to problem now