Mixing
What does $:=$ mean?
$begingroup$
What does $:=$ mean? For example:
Consider the subset $ mathbb{S} = { p in mathbb {P_4} ( mathbb{R,R} ) | P(2)=0 } $
Suppose $p$, $q$ are in $mathbb{S}$, so $p(2)=q(2)=0$. Then $r := p + q$ is also a polynomial of degree at most $4$ and $r(2) = p(2) + q(2)=0+0=0$
Is it just another notation for the $=$ sign? Or is there any significance on having a : in front of it?
notation
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
$endgroup$
add a comment |
$begingroup$
What does $:=$ mean? For example:
Consider the subset $ mathbb{S} = { p in mathbb {P_4} ( mathbb{R,R} ) | P(2)=0 } $
Suppose $p$, $q$ are in $mathbb{S}$, so $p(2)=q(2)=0$. Then $r := p + q$ is also a polynomial of degree at most $4$ and $r(2) = p(2) + q(2)=0+0=0$
Is it just another notation for the $=$ sign? Or is there any significance on having a : in front of it?
notation
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
$endgroup$
add a comment |
$begingroup$
What does $:=$ mean? For example:
Consider the subset $ mathbb{S} = { p in mathbb {P_4} ( mathbb{R,R} ) | P(2)=0 } $
Suppose $p$, $q$ are in $mathbb{S}$, so $p(2)=q(2)=0$. Then $r := p + q$ is also a polynomial of degree at most $4$ and $r(2) = p(2) + q(2)=0+0=0$
Is it just another notation for the $=$ sign? Or is there any significance on having a : in front of it?
notation
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
$endgroup$
What does $:=$ mean? For example:
Consider the subset $ mathbb{S} = { p in mathbb {P_4} ( mathbb{R,R} ) | P(2)=0 } $
Suppose $p$, $q$ are in $mathbb{S}$, so $p(2)=q(2)=0$. Then $r := p + q$ is also a polynomial of degree at most $4$ and $r(2) = p(2) + q(2)=0+0=0$
Is it just another notation for the $=$ sign? Or is there any significance on having a : in front of it?
notation
notation
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
edited Sep 5 '15 at 14:22
Silbol
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
asked Sep 5 '15 at 13:40
SilbolSilbol
3613
3613
add a comment |
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
It usually means: "we are defining what's on the left of := to be wha't on the right". This distinction originates from computer languages, where the mere equality symbol "=" denotes an assignment of one variable's value to another's. For example, in Mathematica they use "==" for being equal, and "=" for assignment.
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
$endgroup$
$begingroup$
"to equal", to be more precise.
$endgroup$
– user236182
Sep 5 '15 at 13:42
$begingroup$
@user236182 could you elaborate on what you mean? I don't think I've ever heard that phrasing.
$endgroup$
– Omnomnomnom
Sep 5 '15 at 13:47
$begingroup$
Does this really originate from computer languages? I've always thought this myself, too (which is why I prefer it). It does seem to be used in place of $equiv$ and $triangleq$ in more modern texts, and it coincides with Maple's assignment operator.
$endgroup$
– Chester
Sep 5 '15 at 13:49
1
$begingroup$
@Chester -- wiki suggests that := originated with ALGOL in 1958 and was popularized by Pascal, see en.wikipedia.org/wiki/Assignment_%28computer_science%29
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:55
$begingroup$
...though I guess that leaves open the question whether := had prior mathematical use
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:56
|
show 4 more comments
$begingroup$
The symbol stands for a definition. Sometimes you can also find $doteq$ or $overset{mathrm{def}}{=}$. However these two symbols are completely symmetric, so that you can't tell $a:=b$ from $a=:b$. The difference is that $b$ is known and $a$ is defined in the first case, the other way round in the second.
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
$endgroup$
add a comment |
$begingroup$
As others have said, the symbol $:=$ means "is defined to be", so $a := b$ means "we define $a$ to be $b$". Other symbols sometimes use include $equiv, stackrel{def}{=}, stackrel{Delta}{=},leftarrow$. In algorithms, this symbol is usually thought of as assigning a value, so that $a := b$ means that we assign the value $b$ to $a$. This is to make clear the destinction between for example
$$x = x + 1$$
and
$$x := x + 1.$$
The first, as a mathematical statement is of course wrong, whereas the second statement simply means that we increase $x$ by one.
Outside of algorithms, the symbol is also used, but here the distinction is more subtle. Here, a statement such as $a := b$ would mean that "$a$ is equal to $b$ because this is how we define it", or simply that we use $a$ as a name for $b$, usually because $b$ is a lengthy expression and we want $a$ to be a more compact symbolism for the same thing. This is in contrast to a statement such as $a = b$, where we say that $a$ and $b$ are equal as a consequence of something else, and not merely because we say so.
Authors who use a symbol like $:=$ to define equalities are very rarely consistent in this use, however, and do not use it every single time they define something, but only when they want to highlight that some relationship holds because it has been defined that way.
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
$endgroup$
add a comment |
$begingroup$
“$a = b$” means “$a$ is equal to $b$” while “a := b” means “let $a$ be equal to $b$”. The pattern can be extended to some other notation: “$a :Leftrightarrow b$” means “let $a$ be equivalent to $b$”, “$x :∈ X$” means “let $x$ be an element of $X$”, “A :⊆ X” means “let $A$ be a subset of $X$”.
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It usually means: "we are defining what's on the left of := to be wha't on the right". This distinction originates from computer languages, where the mere equality symbol "=" denotes an assignment of one variable's value to another's. For example, in Mathematica they use "==" for being equal, and "=" for assignment.
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
$endgroup$
$begingroup$
"to equal", to be more precise.
$endgroup$
– user236182
Sep 5 '15 at 13:42
$begingroup$
@user236182 could you elaborate on what you mean? I don't think I've ever heard that phrasing.
$endgroup$
– Omnomnomnom
Sep 5 '15 at 13:47
$begingroup$
Does this really originate from computer languages? I've always thought this myself, too (which is why I prefer it). It does seem to be used in place of $equiv$ and $triangleq$ in more modern texts, and it coincides with Maple's assignment operator.
$endgroup$
– Chester
Sep 5 '15 at 13:49
1
$begingroup$
@Chester -- wiki suggests that := originated with ALGOL in 1958 and was popularized by Pascal, see en.wikipedia.org/wiki/Assignment_%28computer_science%29
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:55
$begingroup$
...though I guess that leaves open the question whether := had prior mathematical use
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:56
|
show 4 more comments
$begingroup$
It usually means: "we are defining what's on the left of := to be wha't on the right". This distinction originates from computer languages, where the mere equality symbol "=" denotes an assignment of one variable's value to another's. For example, in Mathematica they use "==" for being equal, and "=" for assignment.
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
$endgroup$
$begingroup$
"to equal", to be more precise.
$endgroup$
– user236182
Sep 5 '15 at 13:42
$begingroup$
@user236182 could you elaborate on what you mean? I don't think I've ever heard that phrasing.
$endgroup$
– Omnomnomnom
Sep 5 '15 at 13:47
$begingroup$
Does this really originate from computer languages? I've always thought this myself, too (which is why I prefer it). It does seem to be used in place of $equiv$ and $triangleq$ in more modern texts, and it coincides with Maple's assignment operator.
$endgroup$
– Chester
Sep 5 '15 at 13:49
1
$begingroup$
@Chester -- wiki suggests that := originated with ALGOL in 1958 and was popularized by Pascal, see en.wikipedia.org/wiki/Assignment_%28computer_science%29
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:55
$begingroup$
...though I guess that leaves open the question whether := had prior mathematical use
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:56
|
show 4 more comments
$begingroup$
It usually means: "we are defining what's on the left of := to be wha't on the right". This distinction originates from computer languages, where the mere equality symbol "=" denotes an assignment of one variable's value to another's. For example, in Mathematica they use "==" for being equal, and "=" for assignment.
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
$endgroup$
It usually means: "we are defining what's on the left of := to be wha't on the right". This distinction originates from computer languages, where the mere equality symbol "=" denotes an assignment of one variable's value to another's. For example, in Mathematica they use "==" for being equal, and "=" for assignment.
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
edited Sep 5 '15 at 13:43
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
answered Sep 5 '15 at 13:41
uniquesolutionuniquesolution
8,6061823
8,6061823
$begingroup$
"to equal", to be more precise.
$endgroup$
– user236182
Sep 5 '15 at 13:42
$begingroup$
@user236182 could you elaborate on what you mean? I don't think I've ever heard that phrasing.
$endgroup$
– Omnomnomnom
Sep 5 '15 at 13:47
$begingroup$
Does this really originate from computer languages? I've always thought this myself, too (which is why I prefer it). It does seem to be used in place of $equiv$ and $triangleq$ in more modern texts, and it coincides with Maple's assignment operator.
$endgroup$
– Chester
Sep 5 '15 at 13:49
1
$begingroup$
@Chester -- wiki suggests that := originated with ALGOL in 1958 and was popularized by Pascal, see en.wikipedia.org/wiki/Assignment_%28computer_science%29
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:55
$begingroup$
...though I guess that leaves open the question whether := had prior mathematical use
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:56
|
show 4 more comments
$begingroup$
"to equal", to be more precise.
$endgroup$
– user236182
Sep 5 '15 at 13:42
$begingroup$
@user236182 could you elaborate on what you mean? I don't think I've ever heard that phrasing.
$endgroup$
– Omnomnomnom
Sep 5 '15 at 13:47
$begingroup$
Does this really originate from computer languages? I've always thought this myself, too (which is why I prefer it). It does seem to be used in place of $equiv$ and $triangleq$ in more modern texts, and it coincides with Maple's assignment operator.
$endgroup$
– Chester
Sep 5 '15 at 13:49
1
$begingroup$
@Chester -- wiki suggests that := originated with ALGOL in 1958 and was popularized by Pascal, see en.wikipedia.org/wiki/Assignment_%28computer_science%29
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:55
$begingroup$
...though I guess that leaves open the question whether := had prior mathematical use
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:56
$begingroup$
"to equal", to be more precise.
$endgroup$
– user236182
Sep 5 '15 at 13:42
$begingroup$
"to equal", to be more precise.
$endgroup$
– user236182
Sep 5 '15 at 13:42
$begingroup$
@user236182 could you elaborate on what you mean? I don't think I've ever heard that phrasing.
$endgroup$
– Omnomnomnom
Sep 5 '15 at 13:47
$begingroup$
@user236182 could you elaborate on what you mean? I don't think I've ever heard that phrasing.
$endgroup$
– Omnomnomnom
Sep 5 '15 at 13:47
$begingroup$
Does this really originate from computer languages? I've always thought this myself, too (which is why I prefer it). It does seem to be used in place of $equiv$ and $triangleq$ in more modern texts, and it coincides with Maple's assignment operator.
$endgroup$
– Chester
Sep 5 '15 at 13:49
$begingroup$
Does this really originate from computer languages? I've always thought this myself, too (which is why I prefer it). It does seem to be used in place of $equiv$ and $triangleq$ in more modern texts, and it coincides with Maple's assignment operator.
$endgroup$
– Chester
Sep 5 '15 at 13:49
1
1
$begingroup$
@Chester -- wiki suggests that := originated with ALGOL in 1958 and was popularized by Pascal, see en.wikipedia.org/wiki/Assignment_%28computer_science%29
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:55
$begingroup$
@Chester -- wiki suggests that := originated with ALGOL in 1958 and was popularized by Pascal, see en.wikipedia.org/wiki/Assignment_%28computer_science%29
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:55
$begingroup$
...though I guess that leaves open the question whether := had prior mathematical use
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:56
$begingroup$
...though I guess that leaves open the question whether := had prior mathematical use
$endgroup$
– Gregory J. Puleo
Sep 5 '15 at 13:56
|
show 4 more comments
$begingroup$
The symbol stands for a definition. Sometimes you can also find $doteq$ or $overset{mathrm{def}}{=}$. However these two symbols are completely symmetric, so that you can't tell $a:=b$ from $a=:b$. The difference is that $b$ is known and $a$ is defined in the first case, the other way round in the second.
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
$endgroup$
add a comment |
$begingroup$
The symbol stands for a definition. Sometimes you can also find $doteq$ or $overset{mathrm{def}}{=}$. However these two symbols are completely symmetric, so that you can't tell $a:=b$ from $a=:b$. The difference is that $b$ is known and $a$ is defined in the first case, the other way round in the second.
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
$endgroup$
add a comment |
$begingroup$
The symbol stands for a definition. Sometimes you can also find $doteq$ or $overset{mathrm{def}}{=}$. However these two symbols are completely symmetric, so that you can't tell $a:=b$ from $a=:b$. The difference is that $b$ is known and $a$ is defined in the first case, the other way round in the second.
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
$endgroup$
The symbol stands for a definition. Sometimes you can also find $doteq$ or $overset{mathrm{def}}{=}$. However these two symbols are completely symmetric, so that you can't tell $a:=b$ from $a=:b$. The difference is that $b$ is known and $a$ is defined in the first case, the other way round in the second.
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
answered Sep 5 '15 at 13:51
SiminoreSiminore
30.4k33368
30.4k33368
add a comment |
add a comment |
$begingroup$
As others have said, the symbol $:=$ means "is defined to be", so $a := b$ means "we define $a$ to be $b$". Other symbols sometimes use include $equiv, stackrel{def}{=}, stackrel{Delta}{=},leftarrow$. In algorithms, this symbol is usually thought of as assigning a value, so that $a := b$ means that we assign the value $b$ to $a$. This is to make clear the destinction between for example
$$x = x + 1$$
and
$$x := x + 1.$$
The first, as a mathematical statement is of course wrong, whereas the second statement simply means that we increase $x$ by one.
Outside of algorithms, the symbol is also used, but here the distinction is more subtle. Here, a statement such as $a := b$ would mean that "$a$ is equal to $b$ because this is how we define it", or simply that we use $a$ as a name for $b$, usually because $b$ is a lengthy expression and we want $a$ to be a more compact symbolism for the same thing. This is in contrast to a statement such as $a = b$, where we say that $a$ and $b$ are equal as a consequence of something else, and not merely because we say so.
Authors who use a symbol like $:=$ to define equalities are very rarely consistent in this use, however, and do not use it every single time they define something, but only when they want to highlight that some relationship holds because it has been defined that way.
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
$endgroup$
add a comment |
$begingroup$
As others have said, the symbol $:=$ means "is defined to be", so $a := b$ means "we define $a$ to be $b$". Other symbols sometimes use include $equiv, stackrel{def}{=}, stackrel{Delta}{=},leftarrow$. In algorithms, this symbol is usually thought of as assigning a value, so that $a := b$ means that we assign the value $b$ to $a$. This is to make clear the destinction between for example
$$x = x + 1$$
and
$$x := x + 1.$$
The first, as a mathematical statement is of course wrong, whereas the second statement simply means that we increase $x$ by one.
Outside of algorithms, the symbol is also used, but here the distinction is more subtle. Here, a statement such as $a := b$ would mean that "$a$ is equal to $b$ because this is how we define it", or simply that we use $a$ as a name for $b$, usually because $b$ is a lengthy expression and we want $a$ to be a more compact symbolism for the same thing. This is in contrast to a statement such as $a = b$, where we say that $a$ and $b$ are equal as a consequence of something else, and not merely because we say so.
Authors who use a symbol like $:=$ to define equalities are very rarely consistent in this use, however, and do not use it every single time they define something, but only when they want to highlight that some relationship holds because it has been defined that way.
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
$endgroup$
add a comment |
$begingroup$
As others have said, the symbol $:=$ means "is defined to be", so $a := b$ means "we define $a$ to be $b$". Other symbols sometimes use include $equiv, stackrel{def}{=}, stackrel{Delta}{=},leftarrow$. In algorithms, this symbol is usually thought of as assigning a value, so that $a := b$ means that we assign the value $b$ to $a$. This is to make clear the destinction between for example
$$x = x + 1$$
and
$$x := x + 1.$$
The first, as a mathematical statement is of course wrong, whereas the second statement simply means that we increase $x$ by one.
Outside of algorithms, the symbol is also used, but here the distinction is more subtle. Here, a statement such as $a := b$ would mean that "$a$ is equal to $b$ because this is how we define it", or simply that we use $a$ as a name for $b$, usually because $b$ is a lengthy expression and we want $a$ to be a more compact symbolism for the same thing. This is in contrast to a statement such as $a = b$, where we say that $a$ and $b$ are equal as a consequence of something else, and not merely because we say so.
Authors who use a symbol like $:=$ to define equalities are very rarely consistent in this use, however, and do not use it every single time they define something, but only when they want to highlight that some relationship holds because it has been defined that way.
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
$endgroup$
As others have said, the symbol $:=$ means "is defined to be", so $a := b$ means "we define $a$ to be $b$". Other symbols sometimes use include $equiv, stackrel{def}{=}, stackrel{Delta}{=},leftarrow$. In algorithms, this symbol is usually thought of as assigning a value, so that $a := b$ means that we assign the value $b$ to $a$. This is to make clear the destinction between for example
$$x = x + 1$$
and
$$x := x + 1.$$
The first, as a mathematical statement is of course wrong, whereas the second statement simply means that we increase $x$ by one.
Outside of algorithms, the symbol is also used, but here the distinction is more subtle. Here, a statement such as $a := b$ would mean that "$a$ is equal to $b$ because this is how we define it", or simply that we use $a$ as a name for $b$, usually because $b$ is a lengthy expression and we want $a$ to be a more compact symbolism for the same thing. This is in contrast to a statement such as $a = b$, where we say that $a$ and $b$ are equal as a consequence of something else, and not merely because we say so.
Authors who use a symbol like $:=$ to define equalities are very rarely consistent in this use, however, and do not use it every single time they define something, but only when they want to highlight that some relationship holds because it has been defined that way.
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
edited Sep 24 '15 at 15:39
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
answered Sep 5 '15 at 16:01
mrpmrp
3,79051537
3,79051537
add a comment |
add a comment |
$begingroup$
“$a = b$” means “$a$ is equal to $b$” while “a := b” means “let $a$ be equal to $b$”. The pattern can be extended to some other notation: “$a :Leftrightarrow b$” means “let $a$ be equivalent to $b$”, “$x :∈ X$” means “let $x$ be an element of $X$”, “A :⊆ X” means “let $A$ be a subset of $X$”.
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
$endgroup$
add a comment |
$begingroup$
“$a = b$” means “$a$ is equal to $b$” while “a := b” means “let $a$ be equal to $b$”. The pattern can be extended to some other notation: “$a :Leftrightarrow b$” means “let $a$ be equivalent to $b$”, “$x :∈ X$” means “let $x$ be an element of $X$”, “A :⊆ X” means “let $A$ be a subset of $X$”.
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
$endgroup$
add a comment |
$begingroup$
“$a = b$” means “$a$ is equal to $b$” while “a := b” means “let $a$ be equal to $b$”. The pattern can be extended to some other notation: “$a :Leftrightarrow b$” means “let $a$ be equivalent to $b$”, “$x :∈ X$” means “let $x$ be an element of $X$”, “A :⊆ X” means “let $A$ be a subset of $X$”.
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
$endgroup$
“$a = b$” means “$a$ is equal to $b$” while “a := b” means “let $a$ be equal to $b$”. The pattern can be extended to some other notation: “$a :Leftrightarrow b$” means “let $a$ be equivalent to $b$”, “$x :∈ X$” means “let $x$ be an element of $X$”, “A :⊆ X” means “let $A$ be a subset of $X$”.
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
answered Sep 5 '15 at 16:23
user87690user87690
6,5511825
6,5511825
add a comment |
add a comment |
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