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tahayassen
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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?
There is usually some context around the equation if it's an identity. Sometimes people write this symbol, ##\equiv##, when they write an identity.tahayassen said:If you're just given x^{2}+y^{2}=1, how would you know if it's an equation or an identity?
tahayassen said:Functions are identities, right?
A function is a type of relation so I don't know what you mean by it is an "identity". Perhaps you mean the identity map which is a special case of a function.tahayassen said:Functions are identities, right?
This statement is an identity.Studiot said: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]
This has nothing to do with the equation 1/2 + 1/2 = 1, which is purely a relationship with numbers. If you add additional context, such as that 1 represents 1 sheep, you are moving away from the mathematical meaning.Studiot said:If I cut a sheep in half and gave you both halves would that be identical to a whole sheep?
It's not saying that all (plane) triangles are identical - only that their angles add up to the same value. In the sense that this equation applies to all plane triangles, it is an identity - a statement that is always true.Studiot said: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.
This is incorrect. Conditional equations and identities are different kinds of statements. Inequalities are also statements.disregardthat said:Generally I'd say identities are statements, but equations are not.
You are saying something; namely, that for some number(s), 3 times the square of the number plus the number is the same as 2. I can factor this equation and find the values of x that make it a true statement.disregardthat said: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.
disregardthat said: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.
There is usually some context around the equation if it's an identity
Mark44 said:You are saying something; namely, that for some number(s), 3 times the square of the number plus the number is the same as 2. I can factor this equation and find the values of x that make it a true statement.
Mark44 said: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).
My complaint was not that you provided context, but that what you provided was not related to what was asked in the OP, which was about equations.Studiot said: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
There is usually some context around the equation if it's an identity
and then complain when I provide context!
Studiot said:If you don't like counting sheep how about replacing them with a further geometric example?
Cut a triangle in 'halves'.
Call them half-A and half-B.
Let the cut divide the triangle so that half-A has two sides and half-B three.
Do this to a second triangle and add the Halves-A together or in my parlance substitute half A for half B in the original triangle.
You will now have a quadrilateral!
Mark44 said:You are saying something; namely, that for some number(s), 3 times the square of the number plus the number is the same as 2. I can factor this equation and find the values of x that make it a true statement.
This is a good point, but might be a smidgen on the pedantic side. I would venture to say that most mathematics instructors would treat the qualifier "there exists an x such that..." as being implied. Logicians might quibble at this, but I doubt that mathematics instructors would.disregardthat said:As I explained, that would be if you put the "there exists an x" quantifier in front of the equation. I wouldn't call that part of the equation. Rather, the quantifier is added to it. The equation alone is not a complete mathematical statement, but has uses nevertheless. If you put an "for all x" in front, I'd call it an identity.
I'll go out on a limb and say that most mathematics instructors would call this a conditional equation, which is a kind of statement.disregardthat said: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.
disregardthat said:Mark44 said: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).
What kind of conditions are you talking about here? Statements are either true or false, there are no conditions.
An equation, in a mathematical context, is generally understood to mean a mathematical statement that asserts the equality of two expressions.[1] In modern notation, this is written by placing the expressions on either side of an equals sign (=), for example
$$x + 3 = 5$$
Identities
One use of equations is in mathematical identities, assertions that are true independent of the values of any variables contained within them. For example, for any given value of x it is true that x(x - 1) = x^{2} - x.
However, equations can also be correct for only certain values of the variables.[2] In this case, they can be solved to find the values that satisfy the equality. For example, consider the following.
x^{2} - x = 0.
The equation is true only for two values of x, the solutions of the equation. In this case, the solutions are x = 0 and x = 1.
Many mathematicians reserve the term equation exclusively for the second type, to signify an equality which is not an identity. The distinction between the two concepts can be subtle; for example,
(x + 1)^{2} = x^{2} + 2x + 1
is an identity, while
(x + 1)^{2} = 2x^{2} + 2x + 1
is an equation with solutions x = 0 and x = 1. Whether a statement is meant to be an identity or an equation can usually be determined from its context. In some cases, a distinction is made between the equality sign (=) for an equation and the equivalence symbol (##\equiv##) for an identity.
In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: As the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers.
This is a good point, but might be a smidgen on the pedantic side. I would venture to say that most mathematics instructors would treat the qualifier "there exists an x such that..." as being implied. Logicians might quibble at this, but I doubt that mathematics instructors would.
Best Pokemon said:These definitions are taken from the Oxford Concise Dictionary of Mathematics.
identity: An equation which states that two expressions are
equal for all values of any variables that occur, such as x^{2} -
y^{2} = (x + y)(x - y) and x(x - 1)(x - 2) = x^{3} - 3x^{2} + 2x.
Sometimes the symbolis used instead of = to indicate that a
statement is an identity.
equation: A statement that asserts that two mathematical
expressions are equal in value. If this is true for all values of
the variables involved then it is called an *identity, for
example 3(x – 2) = 3x – 6, and where it is only true for some
values it is called a *conditional equation; for example x^{2} – 2x
–3 = 0 is only true when x = –1 or 3, which are known as the
*roots of the equation.
No, the usual meaning is that the expressions on either side have the same value.tahayassen said:I always thought the ≡ symbol meant "is defined as".
Other texts distinguish between identities and conditional equations by using ##\equiv## vs. =.tahayassen said:But I guess if it's defined as something, then it's also an identity. I think if they used that symbol more, then there would be less confusion for new students about the concept of an identity.
I've seen some texts that borrow notation from the Pascal programming language, :=, to indicate that a term or variable is being defined.
Studiot said:Good point.
Pascal is not the only language to use = as the 'assignment operator'
Tayahassen is studying computing.
You then get such programming statements as
x=x+1
Which makes sense in computing terms.
Yes on both.epenguin said:Mark44 or any other authority, this has long time bugged me, though I live with it. Wouldn't you agree that most math e.g. textbooks writes a sort of slang here? That a lot of the 'equations' really should be identities?
These equations are essentially identities. We are defining f to be the product of g and h, pretty much independent of the value of x (as long as x is in the common domain of g and h, of course). The equation with f' can also be considered an identity.epenguin said:As in any old theorem at random - say if you say
If f(x) = g(x)h(x) then f'(x) = g'(x)h(x) + g(x)h'(x)
you are not wanting to say you have this f, g, and h and then it may happen that for some particular values of x that first equation might be satisfied, in which case the second one is also for those particular values?
I agree. It would probably cut down the confusion.epenguin said:Or reformulating in a single [STRIKE]equation[/STRIKE] [STRIKE]expression[/STRIKE] relationsionship
[g(x)h(x)]' = g'(x)h(x) + g(x)h'(x)
should really be an identity, as maybe more than half the 'equations' in the average textbook should be?
Mark44 said:Where students who are new to this technique have problems is recognizing that the only way two polynomials can be identically equal is when the coefficients of their respective terms are equal.
tahayassen said:If ≡ was replaced with =, wouldn't it also be true that the coefficients of their respective terms are equal?
An identity is a mathematical statement that is always true, regardless of the values of the variables involved. It is an equality that holds true for all values of the variables.
An equation is a mathematical statement that shows the relationship between two or more quantities. It consists of an equality sign and expressions on either side that may contain variables.
An identity is always true, while an equation may or may not be true depending on the values of the variables. Additionally, an identity is an equality between two expressions that are identical, whereas an equation may involve different expressions on either side.
Yes, an identity can also be an equation if the expressions on either side of the equality sign are identical. However, not all equations are identities.
To prove that an equation is an identity, you must show that it holds true for all values of the variables involved. This can be done through algebraic manipulation or by substituting different values for the variables and showing that the equation remains true.