Meanings of Math SymbolsDate: 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 proof?) In English words please, how is each of the above three read? Thanks, 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 Check out our web site! http://mathforum.org/dr.math/ |
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