Identity?

Boston

Right I'm doing some differential equations at the moment, and I same across this definition for a solution.

A function is said to be a solution to a differential equation, over a suitable domain of the independent variable, if it's substitution into the differential equation reduces the differential equation to an identity everywhere within that domain.

I just don't get the use of the word Identity in this context. I thought I understood what it meant for logic but now I'm slightly confused to it's meaning. It really doesn't effect my ability to do the math's, just something that's been annoying me.

ecksor

Identity is one of the many words that's overloaded in maths. Here it just means that you've arrived at an equality expression that is true for all values, like the various trignometric identities you'd have met at leaving cert.

Boston

So it's basically superfluous jargon for pointing out the obvious. In logic just say you had P (P being a proposition) and 1 =P, 1 (or a tautology) was called the identity for the and operator and 0 (or a falsehood) was called the identity for the or operator, this kinda made me think identity related to a specific thing, rather then some form of relationship.

Anyway thanks for confirming that.

ecksor

Well, depending on the equation it mightn't be that obvious, so there's some shorthand for it. A good example would be euler's identity:



The meaning that you met in your logic courses is slightly different and reminds me more of the notion of an 'identity element' I that you'd find in group theory such that A*I = A

Boston

I don't really follow what you're getting at with the identity, what part of that equation is the identity. is an Identity merely an object that when used in conjunction with a specific operator reduces another object to 1 or zero?
I mean the Euler thing, that's just complex numbers in Exponential forum.
e^ jpie = (cos 180 + Jsin180) = -1 + 0 = -1 therefore, e^ jpie +1 = 0. As I said I've no problem with the math's and understanding why something equals something else, its just this concept of an identity. I've spent a year hearing it used in allot of different contexts but as yet no one has given a full account to me of its meaning.

when you say 'identity element' is the equivalent to identity matrix in linear algebra.

ecksor

Originally posted by Boston
I don't really follow what you're getting at with the identity, what part of that equation is the identity.

The equation itself is an identity. You seem to be asking me what the identity element is, which is a different thing.

is an Identity merely an object that when used in conjunction with a specific operator reduces another object to 1 or zero?

This is also talking about an identity element (not sure of the exact terminology in your logic course though). Perhaps a way of putting it is the element that doesn't change any other element under a binary operation.

So, to take your example, A AND true = A

The Rationals under multiplication, A * 1 = A.
The Integers under addition, A + 0 = A

The example of the and operator in the context of group theory is a bit sketchy because it doesn't seem to form a group, but again it is possibly just a terminology difference.

As I said I've no problem with the math's and understanding why something equals something else, its just this concept of an identity. I've spent a year hearing it used in allot of different contexts but as yet no one has given a full account to me of its meaning.

As I said in my first post, the term is overloaded. Different meanings in different contexts.

when you say 'identity element' is the equivalent to identity matrix in linear algebra.

If you're multiplying matrices then the identity matrix is the identity element of that group of matrices, but the identity matrix is understood to be a specifc thing no matter what you're doing with them so you can't really say they're exactly the same thing.

There's a fairly fuzzy connection between the different uses of the word I guess, but they have separate definitions.

Check out:

http://mathworld.wolfram.com/Identity.html
http://mathworld.wolfram.com/IdentityElement.html
http://mathworld.wolfram.com/IdentityMatrix.html

Boston

hmm that has cleared up certain things. I think I'll just have to take it case by case, it does seem a tad overly confusing but you have managed to put some perspective on it, thanks.