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

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• Ad VanderVen
In summary: The symbol '+' is defined to work for pairs so we can use it for our pairs. The symbol '&' is defined to work for our pairs and we can use it for our pairs. The use of '+' and '&' are defined in a way that they are compatible with the definitions of the reals so we are able to use them for real numbers too. So we have defined a new operation for pairs, but we can still use the old one. The old one is the new one for real numbers. In summary, the conversation discusses the meanings of the symbols ##=## and ##\equiv## and their different uses. The equality sign ##=## is used for equal quantities, while the equivalence sign ##\equiv## is used
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?

Last edited:
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.

## 1. What is the meaning of the symbol ##=##?

The symbol ##=## is used in mathematics to represent an equation, indicating that the expressions on either side are equal to each other.

## 2. How is the symbol ##\equiv## different from ##=##?

The symbol ##\equiv## is used to represent an equivalence relationship between two mathematical expressions. This means that the expressions are not only equal, but they also have the same properties and characteristics.

## 3. Can the symbols ##=## and ##\equiv## be used interchangeably?

No, the symbols ##=## and ##\equiv## have different meanings and cannot be used interchangeably. Using the wrong symbol can lead to incorrect mathematical statements and equations.

## 4. In what contexts are the symbols ##=## and ##\equiv## commonly used?

The symbol ##=## is commonly used in basic algebra and arithmetic, while ##\equiv## is used in more advanced mathematics, such as abstract algebra and logic.

## 5. Are there any other symbols that are similar to ##=## and ##\equiv##?

Yes, there are other symbols that represent equality and equivalence, such as ##\sim## and ##\cong##. These symbols have slightly different meanings and are used in specific contexts within mathematics.

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