What does the following notation mean: $\{q(x) : x\in A\}$? I have a definition from my lecture notes, but it makes hardly any sense. Could someone explain it to me, maybe with an example? 
What does the following notation mean: $\{q(x) : x\in A\}$?
 A: It means "the set of values taken by the function $q$ in the set $A$".
For instance, $\{ n^2 : n \in 2\mathbb Z \}$ means the set of squares of even integers.
A: Elements of this set are the values of $q(x)$ for $x\in A$.
For instance if $q(x)=x^2$ and $A=\{1,2,3,4\}$, then $\{q(x):x\in A\}=\{1,4,9,16\}$
A: The brackets $\{...\}$ represent a set. Sets contain mathematical objects. 
For example $\{1, 2, 3\}$ means the set of numbers $1, 2, 3$. 
Inside of sets, a colon ($:$) is often used and you should read this as 'such that'. This gives us a restriction on the mathematical objects we are looking at. 
The set $\{q(x) \}$ is the set of values of $q(x)$, i.e. the outputs of the function $q$. If it is written like this, this generally means range or codomain of $q$. 
However, in your case we have $\{q(x) : x \in A\}$. This is the set of values of $q(x)$ such that $x$ is in $A$. In other words, it only concerns those values of $q(x)$ that are mapped from values of $x$ in $A$. 
An example: 
Let $q(x) = x^2$
Then 


*

*$\{q(x) : x \in \mathbb{R\}}$ is the set of values of $x^2$ obtained from all real $x$

*$\{q(x) : x \in \mathbb{N\}}$ is the set of values of $x^2$ obtained from natural numbers: these are just the square numbers (1, 4, 9, 16, ...)

*$\{q(x) : x \in \mathbb{R_+}\}$ is the set of values of $x^2$ obtained from all real and positive $x$

*Let $A = \{1, 4, 5\}$. Then $\{q(x) : x \in A\}$ is $\{1, 16, 25\}$
