The difference between the symbols and

  • #1
Ad VanderVen
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TL;DR Summary
The difference between the symbols and explained by means of examples.
I have struggled for a long time to understand the difference between the meaning of the concept of 'equality' and 'identity' as represented by the symbols and . Can someone explain it to me and give examples where does apply and does not and vice versa?
 
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  • #2
Equality can refer to being equal at a single specific point, whereas identity means they are equal at every point. just means that and are equal at . It says nothing about them at other values of . Or means that they are equal at some value of to be determined, but nothing about other values of . On the other hand, means that they are equal at every value of .
 
  • #3
Ad VanderVen said:
Summary:: The difference between the symbols and explained by means of examples.

I have struggled for a long time to understand the difference between the meaning of the concept of 'equality' and 'identity' as represented by the symbols [ tex ] = /tex ] and [ tex ] \equiv [ /tex ]. Can someone explain it to me and give examples where does apply and does not and vice versa?
This depends totally on the context or the author. The equality sign is protected for equal quantities.

The use of the equivalence sign is less strictly determined. I normally use it for modular arithmetic like

The numbers here represent an entire equivalence class, namely Here is another notation for

Other uses could be definitions. While I write others may write This is more law than definition, but it only serves for demonstration purposes here.

As you already mentioned, can also mean an identity, e.g. means for all values of and similar for . This appears also as which I personally do not like very much. A notation is rare, but cannot completely ruled out.
 
  • #4
Thank you very much. Your explanation is very clear. Can you also explain the following example:

If en and then

.
 
  • #5
Ad VanderVen said:
Thank you very much. Your explanation is very clear. Can you also explain the following example:

If en and then

.
We have a direct sum here by . If we only write then we identify this element with the element of the sum. Hence is an abbreviation for and stands for: is meant as.

An unlucky abbreviation if you ask me. Maybe it should be read as: The addition in is still part of the addition in (or ) since we can identify by
 
  • #6
In this example , , and are real numbers and is an element of the set of all pairs . Furthermore for the set of all pairs the rule applies:

.

In my previous reply I erroneously wrote:

.
 
  • #7
Ad VanderVen said:
In this example , , and are real numbers and is an element of the set of all pairs . Furthermore for the set of all pairs the rule applies:

.

In my previous reply I erroneously wrote:

.
Your "mistake" makes more sense than the correction. The only justification for the use of '&' is that it demonstrates, that there is a new definition for '+' so it's strictly speaking another operation.
 

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