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.
What does {2x|P(x)} mean?
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0meaning *a priori*, that is, unless the $x$ you are talking about are elements of a structure where "multiply by 2" makes sense: for example, the natural numbers $\mathbb{N}$, the real numbers $\mathbb{R}$, the continuous functions $\mathbb{R}\to \mathbb{R}$, etc. – 2011-07-11
4 Answers
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.
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1@kahen: unless $\{x\colon P(x)\}$ is already a set. – 2011-07-11
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 $x$.
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.
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0@fahad: No, the order in which you check the propositions is not relevant: $\{y:\exists x (y=2x \wedge P(x))\}=\{y: \exists x (P(x) \wedge y=2x\}$. In other words, conjunction is commutative. – 2011-07-11
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 $x > 3, x \in \mathbb R$.
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0Hmm, tired. I actually thought about this deeply for about two minutes and reversed it. And then went to sleep. – 2011-07-11