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Triviality refers to the process of obtaining results from a context or an object with little or no effort. The objects used in these situations have simple topological structures.
- Triviality implies having little worth or importance, according to the Oxford Dictionary.
- This term appears frequently in many situations, such as during interactions with people or when reading a book or article.
- In mathematical society, the term is similar with "proved" that is, any theorem can be considered "trivial" once it is proved to be true.
- It is derived from the Latin word "trivium", which means a lower division of liberal arts.
- In other words, it means something that lacks attention or significance.
- Graph theory, group theory and matrix are some common examples of triviality.
Key Terms: Triviality, Group Theory, Number theory, Deep theorem, Topological spaces, Matrix, Graph Theory, Triviality Proof, Non-Trivial Solution, Vector
Triviality Meaning in Maths
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We use the term triviality to describe a result that requires little to no effort to prove or derive. Nobel Laureate Richard Feynman once said, "A simple theorem is a theorem whose proof has been obtained only once."
- It makes no difference how difficult the proof of that theorem is the first time around.
- A "deep theorem" is a word that can be used to describe the inverse of a trivial theorem.
- Now that we've learned what triviality implies, we're undoubtedly curious about its mathematical relevance.
- It is well known that it is used to describe something that is insignificant or trivial.
- Triviality is used in easy-to-prove theorems in the fields of engineering and mathematics.
- We can deduce that it denotes a lack of focus or even a lack of seriousness.
Trivial bundle, trivial loop, trivial proof, trivial basis, trivial theorem, trivial module, trivial representation, and trivial topology are terms similar to the term triviality.
Examples of TrivialityExample 1: Let X be an unknown vector in linear algebra, then
Example 2: Consider the case when 'n' is an integer number. '1' and 'n' are the two obvious elements of 'n'. These are referred to as "trivial factors." Other factors are referred to as "non trivial factors" if they exist. Example 3: A simple group with only one member or variable is referred to as a "trivial group" in contemporary algebra. "Nontrivial" will be used to describe other complicated groupings. Example 4: When discussing graph theory, a trivial graph is one with only one vertex and no edges.
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Triviality Proof
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The statement of logical implication is the name given to the simple proof in logical or mathematical reasoning. A ----> B can be used to represent the implication.
- It represents the fact that the consequent B is always true, even if the antecedent A's veracity is genuine.
- Let's write the triviality truth table:
| A | B | A → B |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | T |
- True trivially refers to the relationship A → B.
- The proof is known as trivial proof.
Trivial and Non-Trivial Solution
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Trivial Solution is a type of solution that includes equations with a basic structure. Even if they are meaningless, they must be included for the purpose of completeness.
- In basic terms, a trivial solution is a straightforward solution to any problem.
- Nontrivial answers are one step more complex than trivial solutions.
- Finding the solution to nontrivial equations can be a little complicated and demanding.
- So, in general, simple solutions include the number 0, whereas non-zero solutions are considered nontrivial.
Example of Trivial and Non-Trivial SolutionExample: For example, if x+2y is an equation, and the values of x and y are set to zero, the solution will be simple; but, if the x and y variables are set to non-zero, the solution will be nontrivial. |
Things to Remember
- Triviality describes a result that requires little or no effort to prove or derive.
- In mathematics, we define it as a quality of things with simple structures.
- The term trivial refers to obvious notions or entities, such as topological spaces and groups, which have a very simple layout.
- A trivial solution is a straightforward solution to any problem.
- For a differential equation, a trivial solution refers to a zero function, and a non-trivial solution refers to an exponential function.
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Sample Questions
Ques. What does the term “trivial” mean in Mathematics? (1 mark)
Ans. In reality, it usually implies that the individual making the claim is unwilling to explain why it is correct.
Ques.How do you find non-trivial examples? (2 marks)
Ans. It can be said that non-trivial examples have a solution of a linear equation in which the value of at least one variable of the equation is not equal to zero.
Ques. What is the difference between trivial and non-trivial solutions? (2 marks)
Ans. A trivial solution is a straightforward solution to any problem. Because nontrivial answers are one step more complex than trivial solutions, finding the solution to nontrivial equations can be a little complicated and demanding.
Ques. What are the various terminologies used for triviality? (2 marks)
Ans. There are several terminologies associated with triviality, such as trivial topology, trivial proof, trivial representation, trivial theorem, trivial bundle, trivial module, trivial foundation, trivial loop, etc.
Ques. State the non-trivial null space? (2 marks)
Ans. A matrix with a non-trivial null space, i.e. one that does not contain only zeros, is never invertible. Every non-invertible matrix, on the other hand, has a non-trivial null space.
Ques. Where does the word “trivial” come from, and what does it mean? (2 marks)
Ans. The name "trivial" comes from the Latin word "trivium," which refers to the bottom division of liberal arts. Something with a lack of importance or significance is referred to as trivial.
Give three examples that can be termed as trivial.
- Empty set: the set having no or null elements
- Trivial group: the mathematical group containing only the identity element
- Trivial ring: which is one that is defined by a singleton set.
Ques. Explain triviality in Number Theory? (2 marks)
Ans. In number theory, it is typically crucial to discover factors of an integer number N. There are four apparent variables in any number N: ±1 and ±N. These are called "trivial factors". Any additional factor, if it existed, would be considered "non trivial".
Ques. What is the use of Triviality in Mathematics? (2 marks)
Ans. We define triviality in mathematics as a quality of things with simple structures. The term trivial refers to very fundamental and obvious notions or entities, such as topological spaces and groups, which have a very simple layout. Nontrivial is the most basic and straightforward antonym of trivial. In both Mathematics and Engineering, we use it to denote non-obvious assertions and easy-to-prove theorems.
Ques. Suppose there are a total of 20 schools in a large area. And each school is connected to only 4 other schools. Calculate the total number of roads or paths connected to all of them? (3 marks)
Ans. Using graph theory we can consider 20 schools as 20 points or vertices. The degree of each vertex is 4. The road or path connecting schools can be considered as an edge.
From the given data we can calculate the sum total degree of all vertices i..e 20 *4 =800
We know that,
Sum of all degrees of each vertex = 2 * (number of edges)
⇒ 80 = 2 * (number of edges)
⇒ number of edges = 80/2 = 40.
Therefore, the total roads connecting schools is 40.
Ques. There are 10 line segments drawn on a piece of paper. Is it possible that each line segment is exactly connected to 3 others? If yes, then find the sum of degrees of all vertices? (2 marks)
Ans. Given there are a total of 10 line segments, which means a total of 10 vertices. The degree of each vertex is 3 and the total degree is 3 * 10 = 30.
The number of edges can be found from the formula:
Sum of all degrees of each vertex = 2 * (number of edges)
⇒ 30 = 2 * (number of edges)
⇒ 30/2 = (number of edges)
⇒ number of edges = 30/2 = 15
Hence, this graph is possible.
Ques. What is an example of a trivial equation? (2 marks)
Ans. Some examples of trivial equation are as follows:
- It is evident that a set without elements cannot exist, so it is regarded as trivial.
- A trivial ring is one that is utilized for a singleton set.
- Trivial groups are algebraic groups that only have the identity element in them.
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