Real Analysis/Reversibility

Real Analysis Reversibility

In analysis, a common mistake in proof analysis is ensuring that operations are reversible. Reversibility is a seldom discussed topic in mathematics, yet it is integral to build the foundation in order to advance in your mathematical career. To describe it simply,

and to be clear,

These two definitions paints the image of when operations are not reversible; irreversible. Simply, it is when an operation, by using the operation, assumes more of a variable than is given. The usual reason this topic is seldom covered, if you examine the examples provided, is that irreversible operations involve a misuse of an axiom or theorem. These mistakes usually arise when the theorem or problem must be proven without knowing the necessary properties or theorems.

Examples Involving Operators

Early in your mathematical career, the first irreversible operation you may have faced is both the squaring operation and the division operations. We will explore these examples in depth.

Squaring

$x^{2}=|x|^{2}$

Because both a positive value and a negative value, when squared, is the same number, any theorem involving squares must have to contend with the possibility of either a positive or negative value. This means that the solution set when exposing the variable x through square roots increases the solution set by 1, namely by introducing the negative of the given value x. Thus, by default, squaring is an irreversible operation.

For this operation to be reversible, the sign of x must be fixed. if the nature of the question forbids x from being either of the values, the possible solution set becomes fixed, namely since through forbidding a certain range of values, you can isolate the solution set.

Division

$x(x^{2}-1)=0{\text{ and }}(x^{2}-1)=0$

This example is easily conveyed because it has a graphical component, namely through roots. By mentioning roots, it should click that this is how dividing x from the first equation to form the second equation is an irreversible operation.

Examples Involving Inequalities

The more tricky questions regarding reversibility arises when dealing with inequalities. With equations, an irreversible operation is as simple as adding on one side without doing it for the other. However, inequalities generally permit one-sided operations.

"Scope Adjusting"

$a<b{\text{ and }}a<c\neq a<b<c$ It is usually an irreversible action to expand an inequality's scope and then shrink it. A common rule of thumb

For this operation to be reversible, using the variables a < b and a < c ≠ d < c, you must show that a < d.

Exercises

The following exercises will be lighter on mathematical rigor in order to prove.

1. List conditions for which squaring is a reversible operation (Hint: imagine ways to restrict the variable that is squared).