The sentence [beautiful math coming... please be patient] $\,x\in\mathbb{R}\,$ is read differently depending on its context:
[beautiful math coming... please be patient]
$\,x\in\mathbb{R}\,$ (self-standing) |
is read as: | ‘ex is in arr’ or ‘ex is a real number’ |
Recall that $\,\mathbb{R}\,$ represents the set of real numbers. If someone is looking at ‘$\,x\in\mathbb{R}\,$’ as it's being read, then saying ‘ex is in arr’ is shortest and simplest. If not, then saying ‘ex is a real number’ conveys the information more clearly. |
Let [beautiful math coming... please be patient] $\,x\in\mathbb{R}\,$ | is read as: | ‘Let ex be in arr’ or ‘Let ex be a real number’ |
When ‘$\,x\in\mathbb{R}\,$’ appears after the word ‘let’, then the word ‘is’ is dropped, and the word ‘be’ is inserted in its place. Possible memory device: Let it be! |
For all [beautiful math coming... please be patient] $\,x\in\mathbb{R}\,$ | is read as: | ‘For all ex in arr’ or ‘For all real numbers ex’ |
The phrase ‘For all $\,x\in\mathbb{R}\,$’ is always followed by something else, which supplies the verb. For example, you might see: For all $\,x\in\mathbb{R},\ \ x + 2 = 2 + x\,$. Therefore, in this context, the words ‘is’ or ‘be’ are dropped, and nothing is inserted in their place. |
Of course, these same rules apply for similar sentences and contexts.
Recall, for example, that $\,\mathbb{Z}\,$ represents the set of integers.
Thus: