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Meanings of Math Symbols

Date: 06/19/97 at 15:19:22
From: Bob Howell
Subject: More help with symbols

Again, your help has been extremely useful. Now I have run into three 
math symbols with which I need assistance:
     | (the vertical bar)
     : (a colon - does it mean the same thing as the vertical bar?)
     a solid black square at the end of a proof (I am familiar with
       Q.E.D. Is this a more modern way to state Q.E.D. or end the

In English words please, how is each of the above three read?  

Bob Howell

Date: 06/19/97 at 16:46:53
From: Doctor Daniel
Subject: Re: More help with symbols

Hi Bob,

Isn't it frustrating how math sometimes really does start to look like 
it's not only entirely without words, but even without numbers and 
only a combination of bizarrely shaped symbols and Greek letters?

Fortunately, yours are fairly easy:

| is read "such that," as is also :

Both of these are usually found inside set definitions.  For example, 
here's a definition of the set of positive even integers:

   A = {x | x > 0 ^ x/2 is an integer}

Of course, there are even more symbol-driven ways of saying this; 
suppose that E represents the symbol for "is an element of," which 
looks like a lower-case e crossed with an upper-case E.  It's an 
important rule that Z is math-speak for the set of integers. (That 
comes from the German word Zahlen, which means numbers.)  So we could 
also define A like this:

   A = {x | x > 0 ^ x/2 E Z}

We read this as "A is the set of all x such that x is greater than 
zero and x divided by 2 is an integer."

The colon is slightly less common, but does occur occasionally for 
the same reason. Another context where the colon happens is in talking 
about functions. Think of a function as an operation that takes 
objects from one set and maps them to another set. So, for example, 
"Age" is a function from the set of people to the set of integers; it 
maps me to 23, my mother to 50, etc.  A shorthand way of saying this 
in math notation is:

   Age : People -> Z

where -> is an arrow; this is read "Age is a function mapping from the 
set of people to the integers."

Another use for the vertical bar is for absolute value; this is the
distance a number is from zero. It's used just like parenthesis; 
|-1| = 1 while |3.1| = 3.1. So here's a definition of the set of real 
numbers between -1 and 1:

   B = {x : |x| < 1 ^ x is a real number} 

is read: "B is the set of all x such that the absolute value of x is 
less than 1 and x is real."

I don't know when that box started to be so common in math writing, 
but yes, it just means the proof is complete and is much like Q.E.D.

-Doctor Daniel,  The Math Forum
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