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}intersected with{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}intersected with{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}unioned with{x.:.-1.<=.x}. Here is what P or Q. looks like on the number line. 
       __________________________________________________ _ _ _ _ _ _
                   -1                                  5
       ~~~~~~~~~~~~~~~~~~~~~\ /~~~~~~~~~~~~~~~~~~~~~~~~
                  {x.:.x.<.5}unioned with{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.

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