Answer to Question #148106 in Discrete Mathematics for Promise Omiponle

Question #148106
Let R be a relation from A to B. Both sets are finite, with |A|=n and |B|=m. Define the complementary relation "R bar" as follows:

R bar={(a, b)|(a,b)∈R}

Calculate |R bar|.
1
Expert's answer
2020-12-10T13:05:21-0500

By definition



Rˉ={(a,b)(a,b)R}\bar{R}=\left\{\left.(a,b)\right|(a,b)\in R\right\}



This means that the set Rˉ\bar{R} consists of ALL POSSIBLE pairs {(x,y)xA   and   yB}\left\{\left.(x,y)\right| x\in A\,\,\,\text{and}\,\,\,y\in B\right\} .

Since the set AA consists of A=n|A|=n elements, and the set BB consists of B=m|B|=m elements, then the number of possible pairs is



Rˉ=nm\left|\bar{R}\right|=n\cdot m

since the elements for the pair (x,y)(x,y) are selected independently of each other.


ANSWER



Rˉ=mn\left|\bar{R}\right|=mn


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