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To my understanding $\{x|x = 7\} = \emptyset$ means that the number seven is a not allowed value. But I do not understand the meaning of "$x|x$". Can anybody please explain $\{x|x = 7\} = \emptyset$ more in detail.

It's from an introduction book into probability theory. It says:" ... Not possible events are for example $\{x|x = 7\} = \emptyset$ , $\{x|x = 0\} = \emptyset$ , ...". Basis is the Throwing Dice example as e. g. also available on the page http://www.mathsisfun.com/data/probability.html

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  • $\begingroup$ Can you provide some context? Where did you see this and what did it say? Also this question as written is better suited for Math.SE $\endgroup$ – shadowtalker Dec 20 '14 at 14:50
  • $\begingroup$ @HH on CV you can use Tex formatting so you can use it to format your equations. (see more) $\endgroup$ – Tim Dec 20 '14 at 15:10
  • $\begingroup$ @ssdecontrol: It's from a introduction book into probability theory. It says:" ... Not possible events are for example {x|x=7}=∅, {x|x=0}=∅, ...". Basis is the Throwing Dice example as e. g. also available on the page mathsisfun.com/data/probability.html $\endgroup$ – HH. Dec 20 '14 at 15:18
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I guess the bar $|$ means "such that". Therefore "the set of possible outcomes of $x$ such that $x=7$ is an empty set"

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The formalization of the "Throwing a Dice" example involves the construction of a set $\Omega$, known as the sample space, whose elements $\omega$ correspond to each possible result of the experiment. Hence, you can take as your sample space the six elements set $$ \Omega =\{ \bullet, \bullet\bullet, \bullet\bullet\bullet, \bullet\bullet\bullet\bullet, \bullet\bullet\bullet\bullet\bullet, \bullet\bullet\bullet\bullet\bullet\bullet \} \, . $$ A random variable $X$ is a function which maps $\Omega$ to the set of real numbers. For example, you can construct a random variable $X$ which gives you the number of dots in the dice's face which turned up. Here is the function: $$ X(\bullet) = 1 \qquad X(\bullet\bullet) = 2 \qquad X(\bullet\bullet\bullet) = 3 $$ $$ X(\bullet\bullet\bullet\bullet) = 4 \qquad X(\bullet\bullet\bullet\bullet\bullet) = 5 \qquad X(\bullet\bullet\bullet\bullet\bullet\bullet) = 6 \, . $$ For some subset $A$ of the real numbers, the inverse image $X^{-1}(A)$ is the set of elements $\omega\in\Omega$ such that $X(\omega)$ belongs to $A$. Symbolically, $$ X^{-1}(A) = \{ \omega\in\Omega : X(\omega)\in A\} \, . $$ Note that you can consider the inverse image of some set by any function. The function doesn't need to have an inverse.

Consider these examples: $$ X^{-1}(\{5\}) = \{ \bullet\bullet\bullet\bullet\bullet \} $$ $$ X^{-1}(\{1,6\}) = \{ \bullet, \bullet\bullet\bullet\bullet\bullet\bullet \} $$ $$ X^{-1}((-\infty,3]) = \{ \bullet, \bullet\bullet,\bullet\bullet\bullet \} $$ Now consider $X^{-1}(\{7\})$. First, write the formal definition $$ X^{-1}(\{7\}) = \{ \omega\in\Omega : X(\omega)= 7\} \, . $$ Is there any $\omega\in\Omega$ such that $X(\omega)=7$? No. Hence, $$ X^{-1}(\{7\}) = \emptyset. $$

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