What does {2x|P(x)} mean? I am learning sets. I have seen preposition like {2x|P(x)}. I wanted to ask what does 2x mean here? I would be thankful if someone could make it simple to understand. Thanks.
 A: This is known as set-builder notation. Here $\rm\:\{\:x\in S\::\:P(x)\:\}\:$ or $\rm\:\{\:x\in S\ |\ P(x)\:\}\:$ denotes the set of all $\rm\:x\in S\:$ satisfying $\rm\:P(x)\:.\:$ If no universe $\rm\:S\:$ is specified then it defaults to the ambient universe.  
As in your example, the notation is sometimes functionally composed, e.g. in a context where $\rm\:n\:$ denotes an integer, $\rm\:\{\:n^2\::\:2\ |\ n\}\:$ is the set of all integers of the form $\rm\:n^2\:$ that are divisible by $\rm\:2\:$, i.e. the set of all even square integers. Note also the above use of $\::\:$ to avoid a possible clash with $|$ (meaning "divides") in number theory.
A: As it is written, I can guess that $p(x)$ is a rule on $x$ (e.g. $x^2-3x=0$). Then you take the set of all $2x$ (multiply $x$ by $2$) such that the rule holds for $x$.
A: It might depend on what sets you're dealing with, but we can assume that we're just dealing with real numbers for now.
So suppose we are dealing only of subsets of the real numbers, and that P(x) is 'true' whenever x is of the form $4m + 1$ or $1 \mod 4$. Then whatever x satisfy that proposition, you multiply by 2. And that's your set.
So since 1, 5, 9, etc. satisfy P(x), 2, 10, 18, etc. will be the set.
A: It means two times $x$. This could probably written in the context where $x$ is considered to be a number, a vector, an integer/real/complex valued polynomial/function, but not in general set theory though. I think you should just see it as an example. For instance, something like
$$
\{ 2x \,|\, x > 3, x \in \mathbb R \} = \{ x \,|\, x \in \mathbb R, x > 6 \}
$$
could be written and $P(x)$ would be $x > 3, x \in \mathbb R$.
