Otdfbt

Avoiding cliches when writing gods

How hard would it be to convert a glider into an powered electric aircraft?

How to skip replacing first occurrence of a character in each line?

How can drunken, homicidal elves successfully conduct a wild hunt?

Reading two lines in piano

Word for a small burst of laughter that can't be held back

Is there any word or phrase for negative bearing?

Building a road to escape Earth's gravity by making a pyramid on Antartica

Bent spoke design wheels — feasible?

Company did not petition for visa in a timely manner. Is asking me to work from overseas, but wants me to take a paycut

Do any instruments not produce overtones?

Is it a problem that pull requests are approved without any comments

How were concentration and extermination camp guards recruited?

Does the growth of home value benefit from compound interest?

Are there cubesats in GEO?

C SIGINT signal in Linux

Smooth switching between 12v batteries, with toggle switch

Is it possible for people to live in the eye of a permanent hypercane?

Aligning object in a commutative diagram

How can Iron Man's suit withstand this?

Why is quantum entanglement surprising?

Traffic law UK, pedestrians

Adding two lambda-functions in C++

How to make thick Asian sauces?

Congruence, Equal, and Equivalence

Uncertain notation in coding theory bookwhat is ≡ operator equal to in math?Symbol for “if and only if”: $implies$ or $iff$?Origin and usage of $therefore$ and $because$How do I learn all the weird symbols and notations?Why do we use “congruent to” instead of equal to?What's the difference (if any) between writing $(n-1)/2$ and $fracn-12$?Congruence subgroup action notationWhat does an equal sign mean in a parenthesis?What is the difference between “$=$” and “$equiv$”?

1

1

$begingroup$

I know this is very basic problem about math. But sometimes confusing.
What is the difference among

Equal Sign $left(,=,right)$

Congruence Sign (we saw this on number theory) $left(,equiv,right)$

Equivalence Sign $left(,iff,right)$

notation popular-math

share|cite|improve this question

edited May 22 at 5:30

J.-E. Pin

19.1k21755

asked May 19 at 22:04

user516076user516076

1017

$endgroup$

add a comment | 

1

1

$begingroup$

I know this is very basic problem about math. But sometimes confusing.
What is the difference among

Equal Sign $left(,=,right)$

Congruence Sign (we saw this on number theory) $left(,equiv,right)$

Equivalence Sign $left(,iff,right)$

notation popular-math

share|cite|improve this question

edited May 22 at 5:30

J.-E. Pin

19.1k21755

asked May 19 at 22:04

user516076user516076

1017

$endgroup$

add a comment | 

$begingroup$

I know this is very basic problem about math. But sometimes confusing.
What is the difference among

Equal Sign $left(,=,right)$

Congruence Sign (we saw this on number theory) $left(,equiv,right)$

Equivalence Sign $left(,iff,right)$

notation popular-math

share|cite|improve this question

edited May 22 at 5:30

J.-E. Pin

19.1k21755

asked May 19 at 22:04

user516076user516076

1017

$endgroup$

share|cite|improve this question

edited May 22 at 5:30

J.-E. Pin

19.1k21755

asked May 19 at 22:04

user516076user516076

1017

share|cite|improve this question

edited May 22 at 5:30

J.-E. Pin

19.1k21755

asked May 19 at 22:04

user516076user516076

1017

edited May 22 at 5:30

J.-E. Pin

19.1k21755

edited May 22 at 5:30

J.-E. Pin

19.1k21755

asked May 19 at 22:04

user516076user516076

1017

asked May 19 at 22:04

user516076user516076

1017

2 Answers
2

active

oldest

votes

1

$begingroup$

Equals can be generalized to an equivalence relation. This means a relation on a set $S$, $sim$ which satisfies the following properties:

1. 
   $asim a$ for all $ain S$ (Reflexive)


2. If $asim b$, then $b sim a$ (Symmetric)


3. If $a sim b$ and $bsim c$, then $a sim c$ (transitive).

Equals should satisfy those 3 properties.

Congruence goes one step further. It is used to indicate that it preserves some kind of operation on the set. In your case, congruence mod $n$ is indicating that $a pmod n$ times $b pmod n$ is the same thing as $ab pmod n$. So you can exchange what it is equivalent to before doing the operation or after and you get the same thing. It is also congruence under addition.

$Leftrightarrow$ is usually talking about the equivalence of two statements. For instance $a in mathbbZ$ is even if and only if ($Leftrightarrow$) $a=2n$ for some $nin mathbbZ$.

share|cite|improve this answer

edited May 21 at 6:26

YuiTo Cheng

3,33071545

answered May 19 at 22:30

CPMCPM

3,1601023

$endgroup$

add a comment | 

1

$begingroup$

The equal sign between two items mean they are the same. Depending the context this equality is defined or assumed to be understood.

For example if $A$ and $B$ are sets, then $A=B$ means every element of $A$ is an element of $B$ and every element of $B$ is an element of $A$.

On the other hand if $a/b$ and $c/d$ are fractions, then $a/b=c/d$ is defined as $ad=bc$

Congruence sign,$left(,equiv,right)$ comes with a (mod). The definition $aequiv b, pmod n $ is that $b-a$ is divisible by $n$

For example $27equiv 13 pmod 7$

The $iff$ sign is if and only if sign and $piff q$ means $p$ implies $q$ and $q$ implies $p$ where $p$ and $q$ are statements.

share|cite|improve this answer

edited May 19 at 22:27

Bernard

127k743120

answered May 19 at 22:23

Mohammad Riazi-KermaniMohammad Riazi-Kermani

44.7k42163

$endgroup$

add a comment | 

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3232361%2fcongruence-equal-and-equivalence%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

Equals can be generalized to an equivalence relation. This means a relation on a set $S$, $sim$ which satisfies the following properties:

1. 
   $asim a$ for all $ain S$ (Reflexive)


2. If $asim b$, then $b sim a$ (Symmetric)


3. If $a sim b$ and $bsim c$, then $a sim c$ (transitive).

Equals should satisfy those 3 properties.

Congruence goes one step further. It is used to indicate that it preserves some kind of operation on the set. In your case, congruence mod $n$ is indicating that $a pmod n$ times $b pmod n$ is the same thing as $ab pmod n$. So you can exchange what it is equivalent to before doing the operation or after and you get the same thing. It is also congruence under addition.

$Leftrightarrow$ is usually talking about the equivalence of two statements. For instance $a in mathbbZ$ is even if and only if ($Leftrightarrow$) $a=2n$ for some $nin mathbbZ$.

share|cite|improve this answer

edited May 21 at 6:26

YuiTo Cheng

3,33071545

answered May 19 at 22:30

CPMCPM

3,1601023

$endgroup$

add a comment | 

1

$begingroup$

Equals can be generalized to an equivalence relation. This means a relation on a set $S$, $sim$ which satisfies the following properties:

1. 
   $asim a$ for all $ain S$ (Reflexive)


2. If $asim b$, then $b sim a$ (Symmetric)


3. If $a sim b$ and $bsim c$, then $a sim c$ (transitive).

Equals should satisfy those 3 properties.

Congruence goes one step further. It is used to indicate that it preserves some kind of operation on the set. In your case, congruence mod $n$ is indicating that $a pmod n$ times $b pmod n$ is the same thing as $ab pmod n$. So you can exchange what it is equivalent to before doing the operation or after and you get the same thing. It is also congruence under addition.

$Leftrightarrow$ is usually talking about the equivalence of two statements. For instance $a in mathbbZ$ is even if and only if ($Leftrightarrow$) $a=2n$ for some $nin mathbbZ$.

share|cite|improve this answer

edited May 21 at 6:26

YuiTo Cheng

3,33071545

answered May 19 at 22:30

CPMCPM

3,1601023

$endgroup$

add a comment | 

$begingroup$

Equals can be generalized to an equivalence relation. This means a relation on a set $S$, $sim$ which satisfies the following properties:

1. 
   $asim a$ for all $ain S$ (Reflexive)


2. If $asim b$, then $b sim a$ (Symmetric)


3. If $a sim b$ and $bsim c$, then $a sim c$ (transitive).

Equals should satisfy those 3 properties.

Congruence goes one step further. It is used to indicate that it preserves some kind of operation on the set. In your case, congruence mod $n$ is indicating that $a pmod n$ times $b pmod n$ is the same thing as $ab pmod n$. So you can exchange what it is equivalent to before doing the operation or after and you get the same thing. It is also congruence under addition.

$Leftrightarrow$ is usually talking about the equivalence of two statements. For instance $a in mathbbZ$ is even if and only if ($Leftrightarrow$) $a=2n$ for some $nin mathbbZ$.

share|cite|improve this answer

edited May 21 at 6:26

YuiTo Cheng

3,33071545

answered May 19 at 22:30

CPMCPM

3,1601023

$endgroup$

share|cite|improve this answer

edited May 21 at 6:26

YuiTo Cheng

3,33071545

answered May 19 at 22:30

CPMCPM

3,1601023

edited May 21 at 6:26

YuiTo Cheng

3,33071545

edited May 21 at 6:26

YuiTo Cheng

3,33071545

answered May 19 at 22:30

CPMCPM

3,1601023

answered May 19 at 22:30

CPMCPM

3,1601023

1

$begingroup$

The equal sign between two items mean they are the same. Depending the context this equality is defined or assumed to be understood.

For example if $A$ and $B$ are sets, then $A=B$ means every element of $A$ is an element of $B$ and every element of $B$ is an element of $A$.

On the other hand if $a/b$ and $c/d$ are fractions, then $a/b=c/d$ is defined as $ad=bc$

Congruence sign,$left(,equiv,right)$ comes with a (mod). The definition $aequiv b, pmod n $ is that $b-a$ is divisible by $n$

For example $27equiv 13 pmod 7$

The $iff$ sign is if and only if sign and $piff q$ means $p$ implies $q$ and $q$ implies $p$ where $p$ and $q$ are statements.

share|cite|improve this answer

edited May 19 at 22:27

Bernard

127k743120

answered May 19 at 22:23

Mohammad Riazi-KermaniMohammad Riazi-Kermani

44.7k42163

$endgroup$

add a comment | 

1

$begingroup$

The equal sign between two items mean they are the same. Depending the context this equality is defined or assumed to be understood.

For example if $A$ and $B$ are sets, then $A=B$ means every element of $A$ is an element of $B$ and every element of $B$ is an element of $A$.

On the other hand if $a/b$ and $c/d$ are fractions, then $a/b=c/d$ is defined as $ad=bc$

Congruence sign,$left(,equiv,right)$ comes with a (mod). The definition $aequiv b, pmod n $ is that $b-a$ is divisible by $n$

For example $27equiv 13 pmod 7$

The $iff$ sign is if and only if sign and $piff q$ means $p$ implies $q$ and $q$ implies $p$ where $p$ and $q$ are statements.

share|cite|improve this answer

edited May 19 at 22:27

Bernard

127k743120

answered May 19 at 22:23

Mohammad Riazi-KermaniMohammad Riazi-Kermani

44.7k42163

$endgroup$

add a comment | 

$begingroup$

The equal sign between two items mean they are the same. Depending the context this equality is defined or assumed to be understood.

For example if $A$ and $B$ are sets, then $A=B$ means every element of $A$ is an element of $B$ and every element of $B$ is an element of $A$.

On the other hand if $a/b$ and $c/d$ are fractions, then $a/b=c/d$ is defined as $ad=bc$

Congruence sign,$left(,equiv,right)$ comes with a (mod). The definition $aequiv b, pmod n $ is that $b-a$ is divisible by $n$

For example $27equiv 13 pmod 7$

The $iff$ sign is if and only if sign and $piff q$ means $p$ implies $q$ and $q$ implies $p$ where $p$ and $q$ are statements.

share|cite|improve this answer

edited May 19 at 22:27

Bernard

127k743120

answered May 19 at 22:23

Mohammad Riazi-KermaniMohammad Riazi-Kermani

44.7k42163

$endgroup$

share|cite|improve this answer

edited May 19 at 22:27

Bernard

127k743120

answered May 19 at 22:23

Mohammad Riazi-KermaniMohammad Riazi-Kermani

44.7k42163

edited May 19 at 22:27

Bernard

127k743120

edited May 19 at 22:27

Bernard

127k743120

answered May 19 at 22:23

Mohammad Riazi-KermaniMohammad Riazi-Kermani

44.7k42163

answered May 19 at 22:23

Mohammad Riazi-KermaniMohammad Riazi-Kermani

44.7k42163