# Reading Set Notation

Need some basic practice with variables first? Introduction to Variables

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:

(self-standing)

‘ex is in arr’

or

‘ex is a real number’

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 ex be in arr’

or

‘Let ex be a real number’

Possible memory device: Let it be!

‘For all ex in arr’

or

‘For all real numbers ex’

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:

- ‘$\,k\in\mathbb{Z}\,$’ (self-standing) is read as: ‘$\,k\,$ is in zee’ or ‘$\,k\,$ is an integer’
- ‘Let $\,k\in\mathbb{Z}\,$’ is read as: ‘Let $\,k\,$ be in zee’ or ‘Let $\,k\,$ be an integer’
- ‘For all $\,k\in\mathbb{Z}\,$’ is read as: ‘For all $\,k\,$ in zee’ or ‘For all integers $\,k\,$’