There are two primary ways to represent the solution set for all real numbers: using set-builder notation and using interval notation.
Set-Builder Notation
Set-builder notation provides a formal way to define a set based on a specific property. For all real numbers, the set-builder notation is:
{ x | x is a real number }
This is read as "the set of all x such that x is a real number." It effectively includes every possible real number in the set.
Interval Notation
Interval notation offers a concise way to represent a range of numbers. To represent all real numbers, you would use:
(−∞, ∞)
This indicates that the solution set includes all numbers from negative infinity to positive infinity. Parentheses are used because infinity is not a number and cannot be included in the set.
Example and Identity
Consider the equation 1 = 1. This equation is always true, regardless of the value of x. Such an equation is called an identity. The solution set for an identity is all real numbers. Therefore, we can represent its solution set as { x | x is a real number } or (−∞, ∞).
