How To Find The Solution Set

Fix-Builder Notation

How to describe a set past proverb what properties its members have.

A Fix is a collection of things (usually numbers).

Instance: {five, seven, 11} is a set.

But we can also "build" a gear up by describing what is in information technology.

Here is a simple example of fix-builder annotation:

It says "the prepare of all x'due south, such that x is greater than 0" .

In other words whatsoever value greater than 0

Notes:

• The "x" is just a identify-holder, information technology could be anything, such as { q | q > 0 }
• Some people use ":" instead of "|", so they write { x : x > 0 }

Type of Number

It is besides normal to show what type of number x is, similar this:

So it says:

"the set of all 10'south that are a member of the Real Numbers,
such that x is greater than or equal to 3"

In other words "all Existent Numbers from 3 upwards"

There are other ways we could take shown that:

Number Types

We saw (the special symbol for Existent Numbers). Here are the mutual number types:

Natural Numbers	Integers	Rational Numbers	Real Numbers	Imaginary Numbers	Complex Numbers

Instance: { k | k > five }

"the set up of all k's that are a member of the Integers, such that thousand is greater than five"

In other words all integers greater than 5.

This could also be written {half dozen, vii, 8, ... } , so:

{ k | k > 5 } = {6, seven, 8, ... }

Why Utilise It?

When we take a unproblematic gear up similar the integers from ii to 6 we can write:

{two, three, four, five, six}

Just how do nosotros list the Real Numbers in the same interval?

{2, two.1, 2.01, ii.001, 2.0001, ... ???

So instead we say how to build the listing:

{ 10 | x ≥ 2 and 10 ≤ 6 }
Start with all Existent Numbers, so limit them betwixt 2 and vi inclusive.

We can also utilise set builder notation to do other things, like this:

{ x | ten = x² } = {0, ane}
All Real Numbers such that x = 10ⁱⁱ
0 and 1 are the merely cases where x = ten²

Another Example:

Example: ten ≤ 2 or 10 > 3

Set up-Builder Note looks like this:

{ x | x ≤ 2 or x >three }

On the Number Line it looks like:

Using Interval notation it looks like:

(−∞, 2]  U  (iii, +∞)

Nosotros used a "U" to hateful Union (the joining together of 2 sets).

Defining a Domain

Set Builder Note is very useful for defining domains.

In its simplest grade the domain is the set of all the values that go into a part.

The part must piece of work for all values we give it, so it is up to the states to make certain we get the domain right!

Instance: The domain of 1/10

1/ten is undefined at x=0 (considering 1/0 is dividing by nothing).

So we must exclude x=0 from the Domain:

The Domain of 1/x is all the Real Numbers, except 0

We can write this as

Dom(1/x) = {x | x ≠ 0}

Example: The domain of g(ten)=1/(x−one)

one/(x−1) is undefined at x=1, so we must exclude x=1 from the Domain:

The Domain of i/(x−1) is all the Real Numbers, except 1

Using set-architect notation it is written:

Dom( g(x) ) = { 10 | 10 ≠ ane}

Example: The domain of √ten

Is all the Real Numbers from 0 onwards, considering nosotros can't accept the square root of a negative number (unless we use Imaginary Numbers, which we aren't).

Nosotros can write this every bit

Dom(√x) = {x | x ≥ 0}

Example The domain of f(10) = x/(x² − 1)

To avoid dividing past zero we need: x^two - ane ≠ 0

Factor: ten^two - i = (x−1)(x+1)

(ten−1)(x+one) = 0 when ten = 1 or x = −1, which we desire to avoid!

So:

Dom( f(10) ) = {x | x ≠ 1, 10 ≠ −ane}

Source: https://www.mathsisfun.com/sets/set-builder-notation.html

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