Open Sentence, Statement
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?
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 An open sentence is neither true nor false until the variable has been replaced by a value.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." This page doesn't mention that aspect: 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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