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?
On
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.
On
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 $\equiv, \stackrel{\operatorname{def}}{=}, \triangleq,\leftarrow$. 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.
It usually means: "we are defining what's on the left of := to be what's 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.