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How do you distinguish between an identity and an equation? 
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#1
Jan2413, 04:42 PM

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If you're just given x^{2}+y^{2}=1, how would you know if it's an equation or an identity? Functions are identities, right?



#2
Jan2413, 05:03 PM

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It is an equation as soon as there is an "=" Functions are abstract objects that take an input and produce an output. What you have written is not a function, because it would have a multivalued output. One uses the word identity, in two cases. First case: With f(x)=0 can mean either for some x or for all x. In the second case one says f(x) is identically zero and you can use three bars as some kind of super equal sign. Second case you have a complicated term and you want to replace it with another complicated term like [itex]\sin^2 x = 1  \cos^2 x[/itex] this type of equations as replacement rules are sometimes called identities.



#3
Jan2413, 05:44 PM

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If A is identical to B the we may replace A by B in all cases and all situations, without restriction.
If A equals B then we may only substitute B for A, subject to restrictions. For example Distance = Speed x Time is an equation, but not an identity since it is subject to the condition that the speed is constant. 


#4
Jan2413, 06:46 PM

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How do you distinguish between an identity and an equation?
For example, (x  2)^{2} ##\equiv## x^{2}  4x + 4. The equation above is true for any real value of x. An equation that is true only for specific values of the variable is called a conditional equation. Every equation that you're asked to solve is of this type. For example, x^{2}  2x =  4. This equation is true for only one value. 


#5
Jan2413, 09:45 PM

C. Spirit
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#6
Jan2513, 03:44 AM

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Generally I'd say identities are statements, but equations are not.
When we mean sin^2x + cos^2x = 1 as an identity, we're saying something. Namely that this is true for all angles x. But if I were to put 3x^2 + x = 2, i'm not saying anything. It can be an exercise, used in as a part of a mathematical statement or even a definition, but by itself it says nothing. I could say "there is an x such that 3x^2+x=2", but I could also say "solve for x when 3x^2+x=2" or "let x be such that 3x^2+x=2". For identities, you'd always begin your sentence with "for all x: sin^2x+cos^2x = 1", quantifying each variable. 


#7
Jan2513, 04:30 AM

P: 5,462

I think you have to be be very careful using just equations to show the difference between equality and identity. Several of the equations offered are suspect.
For example is the following equation an equality or an identity? [tex]\frac{1}{2} + \frac{1}{2} = 1[/tex] If I cut a sheep in half and gave you both halves would that be identical to a whole sheep? If I cut two sheep in half and gave you the back half of each one would that be the same as a whole sheep or the same as the first situation? Another exmple is the equation The sum of angles of a triangle = 180. This equation is true for all plane triangles but does not make all triangles identical. It is not an identity. Identity is also important without an explicit equation using numbers. For example every equilateral triangle is similar but not identical. Similarity amongst triangles is not an identity. However if the triangles are not only equilateral but have one side equal then they are congruent. This means they are the same whatever their orientation. Congruence is an identity. 


#8
Jan2513, 12:44 PM

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#9
Jan2513, 12:54 PM

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In mathematics, a statement has a logical (i.e., true or false) value. A statement can be always true, true only under certain conditions, or never true (i.e., always false). This equation is an example of a conditional equation, a kind of statement. 


#10
Jan2513, 02:30 PM

P: 5,462

Good evening Mark. Your replies differ from the definitions given in my Collins Reference Dictionary of Mathematics. It has quite a bit to say about identities, the identity element, the identity function etc, offering 7 different mathematical cases of the word identity itself in all.
My congruent triangles corresponds to definition 5a. It is true that a half plus a half can be an identity, and also corresponds to case5a but I gave conditions, I think you are invoking case1 incorrectly, "The property of being (another word for) the same individual". If value is all you are interested in then yes, but identity can be stronger than this. 


#11
Jan2513, 03:02 PM

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Hi Studiot,
I have a mathematics dictionary, but not at hand, so I can't compare. I agree that the word "identity" has different meanings in different contexts (e.g., identity function, identity element, etc.), but the context of the OP was in regard to equations. What I said about identities was specific to mathematical statements in the form of equations or inequalities. 


#12
Jan2513, 03:31 PM

P: 5,462

I understand Tayahassen to be asking the very reasonable question what is the difference between an equality (equation) and an identity that we need to distinguish two different properties. And why (and when) do we use = and [itex] \equiv [/itex].
I have always understood that an identity is a stronger statement in some way and reflects the idea I put forward in post#3 that you can replace one side of an identity with the other without visible effect. It is interesting that you write If you don't like counting sheep how about replacing them with a further geometric example? Cut a triangle in 'halves'. Call them halfA and halfB. Let the cut divide the triangle so that halfA has two sides and halfB three. Do this to a second triangle and add the HalvesA together or in my parlance substitute half A for half B in the original triangle. You will now have a quadrilateral! 


#13
Jan2513, 03:55 PM

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To be concise, in the statement (there exists an x: 2x=1), "there exists an x" is the quantifier, and "2x=1" is the equation. That doesn't make "2x=1" a statement. 


#14
Jan2513, 04:07 PM

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And I'm not convinced that cutting a sheep in half says anything useful about the equation 1/2 + 1/2 = 1. 


#15
Jan2513, 04:28 PM

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#16
Jan2513, 05:49 PM

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Clearly no disagreement of substance here.
But in my view you're talking about "For all x, if x = 1/2, then 2x=1", which is a (true) statement using the equation "2x=1". I don't really know of conditional statements, for me, statements are either true or false. "Conditional statements" seems like no more than a label for uncompleted statements. For the sake of clarity in this context uncompleted statements should not be mixed with proper ones. 


#17
Jan2513, 06:32 PM

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FWIW, here's some of the wiki article on equations (http://en.wikipedia.org/wiki/Equation).
(I added the underline in two passages, below) 


#18
Jan2513, 06:49 PM

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That's an informal article citing informal sources.
But to address their explanation, an equation is only a statement when there are no variables involved (in which case any quantifier is irrelevant). So asserting that x+3 is equal to 5 makes sense only when x is a predefined value, not when it's a variable. It is essential that all variables in an expression are bounded in order for it to be a statement (with a truth value). See http://en.wikipedia.org/wiki/Sentenc...matical_logic) So "1=1", or "1+1 = 2" are statements, but "x=1" is not. If, on the other hand, I say "let x = 3", then suddenly "x=3" is a (true) statement, but that's entirely different. 


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