Set Definition

A set is a group of objects referred to as elements of the set. Sets of numbers are used in basic mathematics such as in geometry and trigonometry to evaluate equations. A mathematical equation is a function that maps elements from an original set to elements in an answer set. The first set is the function's domain. It is the group of numbers for a particular function that is defined by mathematical laws or outlined by definition. When defining a domain the Greek letter epsilon is sometimes used to represent the term "elements of" in the definition. If it is undefined the domain is usually assumed to be the set of all real numbers. The answer set is the function's range. The range is defined as the set of all possible numbers that are answers to the function. A mathematical set cannot be ambiguously defined. A set can contain an infinite number of elements, as long as it is well defined. For example, the set of all real numbers is a well defined set that is infinite. This set is sometimes designated with a capital R. One way of comparing infinite sets is by showing a one to one correspondence between each of the set's elements. This comparison can show one infinite set is larger then another infinite set. Comparing the set of all real numbers which includes all the fractions to the set of all natural numbers (sometimes designated with a capital N) shows the first infinite set is infinitely larger than the second infinite set. The empty set is the one set with no elements at all.

In the mathematical equation x + 1, x plus one, the elements of both the domain set and range set are all real numbers.

In the mathematical equation x + 1 for x greater than five, the domain is defined as the set of all real numbers greater than five, the range is all real numbers greater than six.

In the mathematical equation (0 x x) + 1, zero times x plus one, the domain set is all real numbers and the range set is just the number one, because in this equation for any value x the solution is the number one, because zero times any number is always equal to zero.