There is a multitude of mathematical abbreviations and symbols, from the simple “x” to more involved constructions. One abbreviation you may have encountered is X E R or x∈R, where R is the symbol for “the real numbers”.

Real numbers include all positive integers, negative integers, zero, positive fractions (rational numbers), and irrational numbers.

As you probably know, in mathematics, symbols are very powerful tools to express ideas precisely. If we didn’t have symbols, it wouldn’t be possible for mathematicians to communicate their thoughts so easily and quickly.

However, when using symbols, we always need to take care of what they exactly mean.

This is why I would like to introduce the symbol ∈, which means “is an element of”, today. Many people think that the symbol means “equal to” (≠), and thus represent it as such, but this is where you’ll be making a mistake.

Since we need to be responsible while moving around with mathematical symbols by our side, one should never use them without being clear of their meanings.

**What is a Set?**

In order to understand how these symbols work, we first need to know what a set is.

A set is a collection of objects, which can either be numbers, words, people… anything really.

The way we denote that something belongs in a certain set is by writing its name within curly brackets {}, like this {0,1,2}.

**What Does “X ∈ R” Mean?**

**X ∈ R** doesn’t mean that “x equals to R”, but rather, that “x is an element of the set of real numbers **R**“. The correct way to read this out loud would be “x is in R”.

In the same way, when we write x ∉ R, this means “x is not an element of the set of real numbers”, or more easily read out loud as “x is not in R.”.

The symbol ∉ represents the complete opposite of its sister symbol ∈.

If you have done that already, then it’s really easy to figure out which one of them is being used in each case:

If you see x inside a set, then x belongs to it, but if you see x outside a set, then x doesn’t belong to it.

Thus, when using the symbol, ∈ remember that x is always inside the set; and x is not in this set when using ∉.

Now that we’ve got these two symbols sorted out, it’s time to move on and look at what happens when we combine multiple sets:

x ∈ A∩ B means “x belongs to the intersection of the sets A and B” (which means that x has to be a member of both, A and B).

x ∉ A∩ B means “x doesn’t belong to the intersection of the sets A and B” (which means that x is a member of either set A or set B, but not both).

Notice that we’re using ∈ and ∉ with all their possible meaning. Now we can move on to some more examples:

x ∈/∉ R means “either x is an element of the set of real numbers or x isn’t an element of it”. This sentence uses one negative symbol, so it’s read as “either…or…”. On the other hand, if you were to say,

x ∈/∉ Z this would mean “x is either an element of the set of integers or x isn’t an element of it”. However, in this case, since there’s no negative symbol being used so it would be read as “x is…or…”.

When using the symbols ∈ and ∉ you have to take into consideration whether these two are being used with their positive or negative meaning. In other words, you must decide if they’re being used as a negation operator (the one that means “isn’t”) or not. If you’re not sure which one is being used, just ask yourself: when I use the symbol

∈ am I saying that x belongs to the set? Or rather, am I saying that x doesn’t belong to it? The answer will tell you which one you’re using.

Some people prefer to use the symbols ∈ and ∉ with their positive meanings instead of their standard ones, so they always read them as being the “is an element of” symbol.

But no matter how much you might want these two to mean different things from what they already do, it’s very important that you keep track of which one is being used because it makes a huge difference!

**Bottom Line**

**X E R means that x is a real number, i.e., that x is an element of the set of real numbers R.**