# Interval Notation

#### Casio

##### Member
If an example was written;

h(x) = 1/x and the function h has domain R excluding 0, could somebody please explain how the two open intervals include (- infinity, 0) and (0, infinity)?

#### masters

##### Active member
If an example was written;

h(x) = 1/x and the function h has domain R excluding 0, could somebody please explain how the two open intervals include (- infinity, 0) and (0, infinity)?

Hi Casio,

You already said the domain was all real numbers except 0. To represent that in interval notation you would use the union of the two intervals you have.

$$(- \infty, 0) \cup (0, +\infty)$$

And this says the domain includes all real numbers less than 0 together with all real numbers greater than 0. Zero is excluded in the notation by using paretheses instead of brackets.

#### chisigma

##### Well-known member
An interval is 'open' if it doesn't include the extremes... $- \infty$ isn't a number and 0 isn't included...the same is for 0 and $+\infty$...

Kind regards

$\chi$ $\sigma$

#### Casio

##### Member
Thanks again for all replies, it's my confusion. Because I can see then written in the brackets it was confusing I couldn't understand why they are there?

Although they are included in the brackets, and I can see them there, they are not included, which is what confused me.

#### HallsofIvy

##### Well-known member
MHB Math Helper
In general interval notation, "[" or "]" mean "include this endpoint" while "(" and ")" mean "do not include this endpoint".

[a, b] means "all numbers between a and b and a and b themselves".
In set notation: $$[a, b]= \{ x| a\le x\le b\}$$

(a, b) means "all numbers between a and b not including a and b".
In set notation: $$(a, b)= \{ x| a< x< b\}$$

[a, b) means "all numbers between a and b including a but not b".
In set notation: $$[a, b]= \{ x| a\le x< b\}$$

(a, b] means "all numbers between a and b including b but not a".
In set notation: $$[a, b]= \{ x| a< x\le b\}$$

#### Casio

##### Member
In general interval notation, "[" or "]" mean "include this endpoint" while "(" and ")" mean "do not include this endpoint".

[a, b] means "all numbers between a and b and a and b themselves".
In set notation: $$[a, b]= \{ x| a\le x\le b\}$$

(a, b) means "all numbers between a and b not including a and b".
In set notation: $$(a, b)= \{ x| a< x< b\}$$

[a, b) means "all numbers between a and b including a but not b".
In set notation: $$[a, b]= \{ x| a\le x< b\}$$

(a, b] means "all numbers between a and b including b but not a".
In set notation: $$[a, b]= \{ x| a< x\le b\}$$
Very much appreciated for the help and effort you have put into this thread. May I expand and ask additonal questions in relation to what you have wrote above please.

In your first line of set notation, this I read to mean that the interval is closed.

In your second line of set notation, this I read to mean the interval is open.

In your third line of set notation, this I read to mean the interval is half open or half closed.

In your forth line of set notation, this I read to mean the interval is also open or half closed.

The round bracket being open and the square bracket being closed.

If I am understanding the above correctly, the inequalities when used with real numbers would be used as the domain, and if by example I said;

-1 < x < -3 this would be a closed interval and could be written [-1, -3]

The problem I can't understand at the present is in the use of infinity, whether that be positive or negative infinity and how to correctly interpret it in the use of solutions to questions?

#### Plato

##### Well-known member
MHB Math Helper
The problem I can't understand at the present is in the use of infinity, whether that be positive or negative infinity and how to correctly interpret it in the use of solutions to questions?
One always use $$(-\infty,0]$$ and $$(0.\infty)$$ i.e. with infinity we use ( or ).

$$(-\infty,0]$$ is the set of all real numbers less than or equal to zero.

$$(0,\infty)$$ is the set of all real numbers greater than zero.

Notice how in the first case ] is inclusive and in the second ( is exclusive.

#### HallsofIvy

##### Well-known member
MHB Math Helper
We always use "(" with $$-\infty$$ and ")" with $$\infty$$ because those are not "numbers" in the usual sense. They are only symbols mean "no lower bound" and "no upper bound".