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

[Ad VanderVen]

TL;DR 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 $=$ and $\equiv$. Can someone explain it to me and give examples where $=$ does apply and $\equiv$ does not and vice versa?

 

[FactChecker]

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 $x$.

 

[fresh_42]

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.

 

[Ad VanderVen]

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

$(a,0)+(c,0) = (a+c, 0) \equiv a + c$.

 

[fresh_42]

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 $G\oplus H \ni (a,0).$

 

[Ad VanderVen]

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:

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

 

[fresh_42]

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.