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What are the differences between the followings:

  1. Identity $$ \sin^2(\alpha) + \cos^2(\alpha) = 1 $$

  2. Equation $$ 4x = 16 $$

  3. Equality - $x,y$ are mathematical objects. $$ x = y $$

All of the three examples use the same equality sign but they have a different meaning. How do I differentiate between the three, and what does each one of the examples mean?

Edit: I understand the difference between an equation and an identitiy. I want to understand the difference between an equation and identity and an equality of two mathematical objects. I want to know how to ifferentiate between the three because they all use the same equal sign to mean different things.

We all agree that: $x$ and $y$ are matrices and $x=y$ is different from $4x=16$. But why? How do you know that?

mawaior
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    Interpretation is pretty opinion-based – Vincent Batens Jan 16 '24 at 16:21
  • the interpretation is pretty much the same everywhere. In the first two equations represent both sides of the equations some real number (which is a mathematical object). So they are all equalities of mathematical objects. What it means for mathematical objects to be equal is defined for every class of mathematical objects and can differ. It is always clear from the context what equality means. – Vincent Batens Jan 16 '24 at 16:24
  • Does $x= y$ mean that $x$ and $y$ are the exact same object just different labels for the same thing? @VincentBatens – mawaior Jan 16 '24 at 16:25
  • well no usually people use $\cong$ (isomorphic) for having the same structure, just different labelling and $=$ (equal) for being exactly the same, although people sometimes use $=$ where there should be $\cong$, the point is it doesn't matter, it is clear from the context what is meant – Vincent Batens Jan 16 '24 at 16:30
  • Cf. this article by Barry Mazur – J. W. Tanner Jan 16 '24 at 16:49
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    This is a (closed) repost of another post considered a duplicate of: Is equality the same as identity? if this other post does not answer your question, then please explain why – RyRy the Fly Guy Jan 16 '24 at 16:58
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    Already asked here – Mauro ALLEGRANZA Jan 16 '24 at 16:58
  • It doesn't answer my question. The question doesn't mention $x=y$ when $x,y$ are mathematical objects. The question is about the difference between equation and identities. @RyRytheFlyGuy – mawaior Jan 16 '24 at 18:31
  • $a=b$ is an expressions using two terms, i.e. names. The expression asserts that the two names refer to the same thing (number, set,...). – Mauro ALLEGRANZA Jan 17 '24 at 09:08

1 Answers1

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It's not the "$=$" that's changing meanings, but rather the (context around the) variables. The term "identity" generally indicates that the equality in question has a universal quantifier around it, while a general equality is just asserted to hold for a particular value (or values) of the variable(s) in question.

Here's a bit more detail:

In the first example, there is an implicit universal quantifier: when we refer to the "identity" $\sin^2(x)+\cos^2(x)=1$, we're really referring to the somewhat-more-detailed statement $$\mbox{For every $x$, we have }\sin^2(x)+\cos^2(x)=1.$$ (Even this isn't fully detailed since we should say what sort of thing $x$ can be - e.g. can $x$ be a complex number here? - but since this isn't the main issue I'm sliding past it.) By contrast, when we're solving an equation like "$4x=16$," the idea is that some particular $x$ satisfies this property and we're trying to figure out what it is.

That said, note the way that solving an equation like this plays with the universal quantifier: when we turn $4x=16$ into $x=4$, we can think of this as proving that the statement $$\mbox{For every $x$, we have }[4x=16\iff x=4]$$ is true. This has the same form as the identity above, but with a more complicated "matrix" (this is an old-timey term for "the bit inside the quantifier(s)").

We usually suppress the "For every" bit for ease of writing, but this makes things much trickier for those (like the OP, and myself way back when) who try to figure out what's "really going on" under the surface.

Noah Schweber
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  • Thank you very much for the detailed respone! What about the 3rd example? When a question claims that $x=y$ ($x,y$ are mathematical objects) what does it claim? Does it mean that $x$ and $y$ represent the exact same object? Are $x$ and $y$ just two different names for the same thing? – mawaior Jan 16 '24 at 18:24
  • @mawaior I don't understand how that's any different from "$4x=16$." Both $4x$ and $16$ are mathematical objects; the issue is whether there is an implicit quantification going on. Can you give a concrete example of this sort of usage appearing in a confusing way? – Noah Schweber Jan 16 '24 at 18:45
  • Yes, Let $A$ be an invertible matrix, $x,b$ are vector matrices such that: $Ax=b$ Now I don't understand why we can multiply both sides of the equation by $A^{-1}$. Intuitively I agree with that but I still didn't find a "rule" that allows multiplying both sides of the equation by a matrix, who "says" it is allowed?

    Does the question claim that $Ax$ and $b$ are the same mathematical object?

    As I said I intuitively understand why it is true. Intuition is good but not enough in mathematics.

    Sorry for my bad English, I am not a native speaker of the language.

     – mawaior Jan 16 '24 at 19:04
  • @mawaior "I still didn't find a "rule" that allows multiplying both sides of the equation by a matrix, who "says" it is allowed" See this old answer of mine; basically, it's a core axiom about equality. But to be honest I don't really see how this is an issue with interpreting the expression "$Ax=b$." Given the context you've provided, this is saying that for some specific $A$, $x$, and $b$ of appropriate type the objects $Ax$ and $b$ are the same. – Noah Schweber Jan 16 '24 at 19:09
  • I looked at your answer, What is the name of the axiom "replacing a with b"?

    I really like mathematics and I do understand many topics in calculus, linear algebra and discrete mathematics. But I do have "holes" in my understanding of how things work behind the scenes. In high school I wasn't taught why things work, I was given a method to solve things and that's it, it just works, magic. Now I do want to understand how simple things like solving equations work behind the scenes. I do understand that if $a=b$ you can do any operation on both sides but is there a proof that it works?

     – mawaior Jan 16 '24 at 19:24
  • Why does solving equations work? If I have an equation with no solutions we still do operations as if both sides of the equation are equal. I don't understand why this method of solving equations work.

    $x+1=x$ has no solutions but in order to find that the equation has no solutions you treat it like it has solutions, you add $-x$ to both sides and get a contradiction. But why could we treat the equation as if it has solutions?

     – mawaior Jan 16 '24 at 19:27
  • Now I tried filling the "holes" of how things work, but I didn't find other explanations of how it works. The explanations I found were that if $a=b$ we can do any operation because $a$ is equal to $b$ but you now said that $a$ is now always equal to $b$, so why can we do these operations on both sides of the equation? We can do these operations if we know that $a$ is actually equal to $b$. – mawaior Jan 16 '24 at 19:30
  • @mawaior Think of it as a proof by contradiction (technically it's a proof of negation rather than a proof by contradiction, but meh): assuming (say) $x=x+1$ for some particular $x$ we're able to deduce something we know is false, namely $0=1$, and so our initial assumption had to be wrong. This isn't a vicissitude of equality, it's just a general aspect of conditional reasoning in mathematics. – Noah Schweber Jan 16 '24 at 20:34