Negating Statements

Date: 10/27/98 at 01:08:08
From: Heather
Subject: Advanced pre-algebra

What is negation? My math teacher gave me some problems on it: 
"4 + 3 * 5= 35" and "Violins are members of the string family." 

I've asked my parents about it and they don't know. 
Thanks.

Date: 10/30/98 at 01:28:46
From: Doctor Teeple
Subject: Re: Advanced pre-algebra

Hi Heather,

To negate a statement, you write the opposite of what the statement 
says. But before we talk about the opposite of a statement, let's talk 
about the statements themselves. 

A statement is pretty much what it sounds like it should be. It's an 
equation or sentence or a declaration of some sort. It doesn't matter 
whether the statement is true or false; we still consider it to be a 
statement. For example, I could say, "The sky is purple" or "The earth 
is flat." Both of those are statements. I could say, "The U.S. is in 
North America" or "Giraffes are not short." Those are also statements. 

We can negate each of these statements by writing the opposite of what 
it says. So for example, the negation of "The sky is purple" is "The 
sky is not purple." The negation of "Giraffes are not short" is 
"Giraffes are short."

We make statements and negate them without judging whether they are 
true or false. That is another issue. But once we are given that a 
statement is true or false, we can note what happens to the statement 
when we negate it. For example, suppose we know the following: 

   "The sky is purple."               False
   "Giraffes are not short."          True

We negated these and got the following:

   "The sky is not purple."           True     
   "Giraffes are short."              False 

Notice what happened. Negation turns a true statement into a false 
statement and a false statement into a true statement. 

Now, all of the statements we have been working with are sentences. 
We can also do this with math equations. Here are some statements:

   6 * 3 = 18
   4 + 3*2 > 15
   15/7 = 12
   12 + 1 <= 13   (where <= means less than or equal to)            

Some of them are true and some are false, but that is a side issue; we 
can negate them either way. So here are the negations of the above 
statements:

   6 * 3 =/ 18     (where =/ means not equal to)
   4 + 3*2 <= 15
   15/7 =/ 12
   12 + 1 > 13

That's all there is to negating statements. 

I want to warn you to be on the watch for statements that contain words 
like "for every," "for all," or "there exists." Negations of these 
types of statements can be tricky. Here's an entry in the Dr. Math 
archives that might help:

   http://mathforum.org/dr.math/problems/thomas.8.3.96.html   

Please write back if you need more help on those tricky negations or 
more explanation on anything I've said above.

- Doctor Teeple, The Math Forum
  http://mathforum.org/dr.math/