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### Open Sentence, Statement

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Date: 09/18/2001 at 21:24:40
From: Kristin
Subject: Math vocabulary - open sentence and statement

A statement is a mathematical sentence that is either true or false
but not both. An open sentence is a mathematical sentence that has a
variable.

My math teacher told me that y+x = x+y is a statement because it is
always true, but I thought it was an open sentence because it has a
variable. Then she siad x+0 = x was an open sentence, but that is
always true, so wouldn't it be a statement?
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Date: 09/18/2001 at 23:24:56
From: Doctor Peterson
Subject: Re: Math vocabulary - open sentence and statement

Hi, Kristin.

I think your definitions really come down to the question whether you
can tell by looking at it whether the equation is true or false - not
just whether there is a variable involved. It is a statement whether
it is always true or always false, and "open" if its truth depends on
the choice of a variable.

So y+x = x+y, since it is always true (we really call this an
"identity"), would fall in the category of "statement" as you have
defined it. So would x+0 = x; I can see no way to define the terms
that would distinguish these equations. The other possibility would be
that both equations are BOTH statements and open sentences, if "open"
does mean merely that there is a variable in it, as you suggested.

Having said that, I did a web search to check for a clear definition
of the terms. I found this:

Open Sentences - Glencoe/McGraw-Hill Parent and Student Study
Guide, Algebra 1
http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_01/pdf/0105.pdf

Mathematical statements with one or more variables are called open
sentences. An open sentence is neither true nor false until the
variable has been replaced by a value. Finding a replacement for
the variable that results in a true sentence is called solving the
open sentence.

This is not exactly definitive, since although it defines an open
sentence as one with a variable, it assumes that it can't be always
true. So where does an identity fit?

Here's another with the same fault:

Math Vocabulary - John S. Pitonyak
http://wcvt.com/~tiggr/vocab.html

Open sentence
A statement that contains at least one unknown. It becomes true
or false when a quantity is substituted for the unknown. (e.g.,
x + 5 = 9, y - 2 = 7).

Here's yet another, but this time the meaning of "open" is made more
explicit:

Mathematics Glossary N-Z
http://www.blc.edu/fac/rbuelow/MPS/glossaryn-z.htm

Open Sentence
Sentences (equations) that have variables to be replaced. Open
sentences cannot be labeled as true or false, their status is
"open."

Algebra Glossary of Terms
http://www.wtvl.net/honda/glossaryal.htm

open sentence: (10)  A sentence containing one or more variables.

But this one mentions only the "open" concept:

Linear Equations - Vernon McBride
http://www.usd.edu/~vmcbride/CollegeAlgebra/tsld004.htm

Whether an equation is true or false depends on the value
substituted for the variable, and so is called an open sentence.
The equation is neither true nor false until a value is chosen
for the variable.

On the whole, I think I'm right that the primary meaning of "open" is
that no truth value can be assigned, so an identity is not properly an
open sentence. But I wish educators were mathematically trained so
they could give a decent definition of their terms, and be consistent
in their usage! Without clear definitions, it just isn't mathematics.

After thinking about it, I realized that the term "open sentence" is
in fact used by mathematicians, though in an area of math that I don't
know well. It's not quite the same usage as at your level, but it
probably is the source of the concept. This will go well beyond what
you wanted to know, but it may be interesting.

In mathematical logic, an open sentence is defined as one that
contains "free variables," variables that are not "bound" by being
used in a phrase like "for all x" or "there exists an x such that..."
(called a quantifier). You could think of them simply as variables
that have not been given a specific meaning. If we say, "for all x and
y, x+y = y+x", then x and y are bound variables and the sentence is
not open. But if we just say "x+y = y+x" without saying anything about
the variables, then they are presumably free and the sentence would be
considered open. So although the primary meaning of "open" is "not
having a definite truth value, because it depends on the unknown
values of the variables," the fact that the equation happens to be
an identity and is always true doesn't really change the fact that the
variables are unspecified and could conceivably affect the validity of
the sentence.

So if we take the question this way, it would seem that identities,
when stated without explicitly saying that they are always true, might
well be considered open sentences. I'm not sure we should go this far,
though, because the definitions used at your level don't mention
quantifiers at all. I still think the concept has been stretched a
little too far by educators, and is not defined well enough when
removed from its home in formal logic.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra
High School Logic

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