If you have two statement P and Q where P is "x.<.5" and Q is "-2..x".

The statement P and Q. is the set of numbers which satisfies P and Q at the same time. When you write it in set notation the and corresponds to intersection. It would be written as {x.:.x.<.5.and.-1..x} which is the same as {x.:.x.<.5}{x.:.-1..x}. Let's use the number line to visualize this.

```
_______________________________________________o_ _ _ _ _ _ _
{x : x < 5}                  5

_ _ _ _ _ _ _________________________________________________
-1            {x : -1  x}

_ _ _ _ _ _ ____________________________________ _ _ _ _ _ _
-1     {x.:.x.<.5}{x.:.-1..x}    5

```
Now the statement P or Q. means the set of numbers which satisfies either P or Q or both of the statements. When it is written in set notation the or corresponds to union. So P or Q. would be written as {x.:.x.<.5.or.-1..x} which is the same as {x.:.x.<.5}{x.:.-1..x}. Here is what P or Q. looks like on the number line.
```       ________________________________________________ _ _ _ _ _ _
-1                                  5
~~~~~~~~~~~~~~~~~~~~~\ /~~~~~~~~~~~~~~~~~~~~~~~~
{x.:.x.<.5}{x.:.-1..x}

```
Here is another example:

Let P be the statement x < 0 and Q be the statement x.>.100. Let's start with P or Q. Using the number line we can see:

```       ______________o_ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _
{x : x < 0}   0

_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _o________________
100   {x : x > 100}

______________o_ _ _ _ _ _ _ _ _ _ _ _ _ _ _o________________
\                                /
\                              /
{x : x < 0 or x > 100}```

Now if we were to try and do P and Q. the number line picture would be:

```       _______________o___________________________o________________
0                          100 ```
The set of P and Q. is empty since there is no number that can be both less than 0 and greater than 100 at the same time.

As you can see and. and or. are quite different so make sure that you know the difference between them before continuing.