Triviality

Triviality is used to describe a result that needs very less or no effort to prove or derive it. Its synonyms are unimportance, insignificance, in-consequence, etc. Richard Feynman, Nobel prize winner, stated- “a trivial theorem is a theorem whose proof has been obtained once”. It does not matter how difficult the proof of that theorem is in the first time. A “deep theorem” is termed as the opposite of a trivial theorem. In the following article, we are going to discuss the concept of triviality in Mathematics.

The word “trivial” is seen very often in our day to day life. Trivial usually refers to something less significant or is of little value. This word is derived from the Latin word “trivium”, which is used to refer to the lower division of liberal arts. Triviality means something with a lack of attention or lack of seriousness or significance.

Triviality Meaning in Maths

In Mathematics, triviality is a property of objects having simple structures. The word trivial is used for simple and evident concepts or things, such as – topological spaces and groups that have a simple arrangement. The antonym of trivial is nontrivial. It is used to indicate non-obvious statements and easy-to-prove theorems in Mathematics as well as in Engineering. Thus, the triviality shows a simple aspect of a definition or proof.

Examples of Triviality

  • An empty set is trivial since it is apparent that it contains no elements.
  • A trivial ring is a ring that is used for a singleton set.
  • A trivial group is an algebraic group which involves only the identity element.

There are few more terms related to triviality, for example – trivial bundle, trivial basis, trivial loop, trivial proof, trivial theorem, trivial representation, trivial module, trivial topology, etc.

Triviality Proof

In logical or mathematical reasoning, the trivial proof is the statement of logical implication. The implication is denoted by A → B.It indicates that consequent B is always true, irrespective of the truth of the antecedent A. Let us write the truth table for this:

A

B

A → B

T

T

T

T

F

T

F

T

T

F

F

T

The relation A → B is called is true trivially. Its proof is referred to as trivial proof.

Trivial and Nontrivial Solutions

Trivial solutions are the solutions to some equations which have a simple structure. They are of less importance but cannot be skipped due to the sake of completeness. In other words, a simple solution to an equation is termed a trivial solution. Nontrivial solutions are a little more difficult to find than trivial ones. So basically, it is said that trivial solutions involve number 0 and non-zero solutions are said to be nontrivial.

Example:

If x+2y is the equation, then putting the value of x and y equal to zero, then the solution will be trivial, but if we put non-zero values to x and y variables, the solution will be nontrivial.

Examples of Triviality

  1. In linear algebra, let X be the unknown vector and A be is the matrix and O is zero vector. One simple solution of matrix equation AX = O is X = 0 which is known as “trivial solution”. Any other non-zero solution is termed as a “nontrivial” solution.
  2. Let us assume that ‘n’ be an integer number. The two certain factors of ‘n’ are ‘1’ and ‘n’. These are called “trivial factors“. If there are other factors, they will be called “nontrivial factors”.
  3. In modern algebra, the simple group with just one member or variable in it is called “trivial group“. Other complex groups are called “nontrivial”.
  4. When we discuss graph theory, the trivial graph is a graph having just one vertex and no edges.