Question

What is R* in Maths?

Answer 1

Hint: A notation is a set of visuals or symbols, letters, and shortened formulations that are used to represent technical facts and numbers by convention in creative and scientific professions. As a result, a notation is also a group of connected symbols with arbitrary meanings that are used to assist organized communication within a subject of expertise or field of study.

Complete step-by-step solution:
Case 1: The reflexive-transitive closure of a relation R in a set X, is the smallest reflexive and transitive relation containing R and it is equal to the union of the diagonal relation \[\Delta \_X\] and all the positive powers of R, where the \[{m_{th}}\] power \[{R^m}\] is the composition \[R \times R \times ... \times R\] (m copies of R). Thus for a, b in X, \[\left( {a,b} \right) \in {R^*}\] if and only if either a=b, or there are elements \[{x_1},{x_2},...,{x_k}\], where k is a positive integer, \[\left( {a,{x_1}} \right),\left( {a,{x_2}} \right),...,\left( {a,{x_k}} \right)\], all belong to the relation R.
Then if we have (a,b) and (b,c) in R, we also put (a,c), if it’s not already there to get \[{R^2}\], and then repeat to get \[{R^3}\], and after a finite number of steps nothing new will be added, so the resulting relation \[R^*\] is the reflexive transitive closure of R.

Case 2: We consider a set of real numbers not including zero. Now we consider the multiplication operation for those numbers. The multiplication of different sets of numbers will be \[{R^1},{R^2},...,{R^*}\], which is also denoted by R*.
So R* in maths is reflexive transitive closure and a set of non-zero real numbers.

Note: For reflexive transitive closure take the union of only diagonal entries. For a set of real numbers, a real number should not be zero otherwise the overall multiplication will become zero.