Universal Set

The universal set is the set of all the elements acceptable to a particular mathematical function. The universal set contains all possible numbers that work with the function. Normally, the universal set is the set of real numbers. It is equivalent to the unrestricted domain of the function. The universal set for the function is restricted from any number that yields an impossible result, such as an imaginary or infinite number. The square root function restricts the use of negative numbers, so its universal set is the set of all real numbers greater or equal to zero. A complement to a set is the difference between the original set and the universal set. It is a proper subset of the universal set. A proper subset is any set contained entirely within the universal set, but is not equivalent to it. Every element must be in the universal set. The power set is a special set that contains all the proper subsets of a set. A proper subset can have an infinite number of elements. For example, the set of integers is a well-defined infinite set that is a proper subset of the set of real numbers which is also infinite.

In the mathematical equation x + 1 the universal set is the set of all real numbers.

In the mathematical equation the square root of x the function restricts the universal set to the set of all real numbers greater than zero, because the square root of a negative number is undefined or an imaginary number.