The Invariant principle — Introduction and examples

Understand the famous mathematical property which acted as bane for several complex real life problems and still kicks a punch here and there.

Introduction

The dictionary defines ‘invariant’ as ‘never changing’, and well that’s really all there is. Invariant in mathematics, is a property held by a mathematical object, which remains same even after repetitive transformation of the object. If for some objects that property is different, then we can never reach from the original object to the newer ones, by trying the same transformations. This may sound tricky, but its really helpful and in some cases may even solve the problem.

Types

There are basically two broad categories,

1. Invariance: Property that stays constants.
2. Mono-variance: Property that changes in only one-direction. Its either always increasing or decreasing.

As Arthur Engel said it, “Invariance principle is a heuristic principle, its best learned by experience”, lets try to solve some examples to understand.

Example 1: Lone survivor