What does the following notation mean: $\{q(x) : x\in A\}$?

Question

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: $\{q(x) : x\in A\}$?

Answer 1

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.

Answer 2

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 $\{q(x):x\in A\}=\{1,4,9,16\}$

Answer 3

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 $\{1, 16, 25\}$