B The difference between the symbols ##=## and ##\equiv##

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The difference between the symbols ##=## and ##\equiv## 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 ##\equiv##. Can someone explain it to me and give examples where ##=## does apply and ##\equiv## does not and vice versa?
 
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Equality can refer to being equal at a single specific point, whereas identity means they are equal at every point. ##f(x_0) = h(x_0)## just means that ##f## and ##h## are equal at ##x_0##. It says nothing about them at other values of ##x##. Or ##f(x) = h(x)## means that they are equal at some value of ##x## to be determined, but nothing about other values of ##x##. On the other hand, ##f(x) \equiv h(x)## means that they are equal at every value of ##x##.
 
Ad VanderVen said:
Summary:: The difference between the symbols ##=## and ##\equiv## 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 ##\equiv## 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
$$
2\equiv 9 \mod 7 \Longleftrightarrow 7\,|\,(9-2)
$$
The numbers here represent an entire equivalence class, namely ##2=[2]_7=\{\ldots,-5,2,9,16,\ldots\}.## Here ##2\equiv 9 \mod 7## is another notation for ##[2]_7=[9]_7.##

Other uses could be definitions. While I write ##R:=U/I## others may write ##R\equiv U/I.## This is more law than definition, but it only serves for demonstration purposes here.

As you already mentioned, ##'\equiv'## can also mean an identity, e.g. ##f\equiv 1## means ##f(x)=1## for all values of ##x## and similar ##f\equiv 0## for ##f(x)=0##. This appears also as ##f(x)\equiv 0## which I personally do not like very much. A notation ##f(x)\equiv x^2## is rare, but cannot completely ruled out.
 
Thank you very much. Your explanation is very clear. Can you also explain the following example:

If ##a \equiv (a, 0)## en ##c \equiv (c, 0)## and ##(a,b)+(c,d) = (a+c, b+d)## then

##(a,0)+(c,0) = (a+c, 0) \equiv a + c##.
 
Ad VanderVen said:
Thank you very much. Your explanation is very clear. Can you also explain the following example:

If ##a \equiv (a, 0)## en ##c \equiv (c, 0)## and ##(a,b)+(c,d) = (a+c, b+d)## then

##(a,0)+(c,0) = (a+c, 0) \equiv a + c##.
We have a direct sum here by ##(a,b)+(c,d) = (a+c, b+d)##. If we only write ##a## then we identify this element with the element ##(a,0)## of the sum. Hence ##a## is an abbreviation for ##(a,0)## and ##'\equiv'## stands for: is meant as.

An unlucky abbreviation if you ask me. Maybe it should be read as: The addition in ##G## is still part of the addition in ##G\oplus H## (or ##G\times H##) since we can identify ##G\ni a## by ##G\oplus H \ni (a,0).##
 
In this example ##a##, ##b##, ##c## and ##d## are real numbers and ##(a,b)## is an element of the set of all pairs ##(a,b)##. Furthermore for the set of all pairs ##(a,b)## the rule applies:

##(a,b) \, \& \, (c,d) = (a+c, b+d)##.

In my previous reply I erroneously wrote:

##(a,b) + (c,d) = (a+c,b+d)##.
 
Ad VanderVen said:
In this example ##a##, ##b##, ##c## and ##d## are real numbers and ##(a,b)## is an element of the set of all pairs ##(a,b)##. Furthermore for the set of all pairs ##(a,b)## the rule applies:

##(a,b) \, \& \, (c,d) = (a+c, b+d)##.

In my previous reply I erroneously wrote:

##(a,b) + (c,d) = (a+c,b+d)##.
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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