For All Notation

The (for all) symbol is used in math to describe the meaning of one or more variables in a statement. Typically, the symbol is used in an expression like this:

In plain language, this expression means “for all in the set of real numbers”. This type of expression is usually followed by another statement that should be able to be proven true or false.

  • The symbol (element of) represents set membership.
  • The symbol (reals) represents the set of real numbers.


For example, the two expressions below form a statement where the “for all” notation is used to describe the meaning of the two variables and . This is followed by a verifiable statement using the variables.

These two expressions can be read as “for all and belonging to the set of real numbers, this equation is true”.

The equation represents the claim which, in this case, is the square of the sum of two numbers. This claim either holds true for every possible combination of two numbers belonging to the set of real numbers or it doesn’t.

As a consequence of the for all notation – if you can find any combination of two numbers in the set of real numbers where the statement is false, then the whole claim is false.