Question #146311

Use Mathematical Induction to show that if MR is the bit matrix representing the relation R, then M^[n]R is the matrix representing R^n. (This was how the question was stated. If you're confused about the terms M^[n]R and R^n, they aren't exponentials, the [n] in the first term is meant to be a superscript and the R a subscript. The n in the second term is a superscript.)


1
Expert's answer
2020-12-02T18:45:28-0500

If RR is a binary relation on a finite set AA , that is RA×AR ⊆ A×A, then RR can be represented by the logical matrix MRM_R whose row and column indices index the elements of AA such that the entries of MRM_R are defined by:


mi,j={1(xi,yj)R0(xi,yj)∉R{\displaystyle m_{i,j}={\begin{cases}1&(x_{i},y_{j})\in R\\0&(x_{i},y_{j})\not \in R\end{cases}}}


Since RR be a symmetric relation on a finite set AA(x,y)R(x,y)\in R implies (y,x)R(y,x)\in R, and therefore mi,j=1m_{i,j}=1 if and only if mj,i=1m_{j,i}=1. It follows that MRM_R is necessarily a symmetric matrix.


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