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The definition of the union set between two sets is: $$A \cup B = \{x | x\in A \lor x\in B\}$$ How do you say this in English? Do you say '$A \cup B$ is the set of all elements $x$, such that $x$ is an element of $A$ or $x$ is an element of $B$'? Isn't this the same as saying $\forall x ( x\in A \lor x \in B)$.

What is the difference between the two?

pregunton
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Carefullcars
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  • You're translation into English is correct. $\forall x(x\in A \vee x \in B)$ is a statement that tells you that every $x$ is in $A$ or in $B$ while ${x\mid x\in A \vee x \in B}$ is the set of all of the $x$ that happen to be in $A$ or in $B$. Do you see the difference? –  Oct 11 '15 at 12:15
  • What has ‘belongs to’ have to see with the euro symbol ;o) – Bernard Oct 11 '15 at 12:16
  • The set notation and the logical sentence are not the same. The first is describing a set and the second is saying that all $x$ are contained in the union of $A$ and $B$. The commonality is in that they involve the union. – Matt Samuel Oct 11 '15 at 12:42
  • you're translation works perfectly, by the way I know your dilemma and in a case like this it has been proved helpful to consult for example the English wiki source like union of sets...@Bernard: I guess in this case the union doesn't consider to kick the $x$ out – user190080 Oct 11 '15 at 12:42

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