by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

Want to say that some element is in a particular set?
Then, you'll use a mathematical sentence similar to ‘$\,x\in\Bbb R\,$’.

Recall that $\,\Bbb R\,$ represents the set of real numbers.
If $\,x\,$ is a real number, then ‘$\,x\in\Bbb R\,$’ is true.
If $\,x\,$ is not a real number, then ‘$\,x\in\Bbb R\,$’ is false.

The sentence   ‘$\,x\in\mathbb{R}\,$’   is read differently depending on its context:

is read as: ‘ex is in arr’
‘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 $\,x\in\mathbb{R}\,$ is read as: ‘Let ex be in arr’
‘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 $\,x\in\mathbb{R}\,$ is read as: ‘For all ex in arr’
‘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.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Going from a Sequence of Operations to an Expression

Consider the sentence:
The best way to read $\,x\in\mathbb{R}\,$ in the previous sentence is:

(an even number, please)