Compositions of Sets 2

Question

Given set A = {a, b, c} relation R = {(a,b),(b,c),(c,a)} relation S = {(a,c),(c,a)}

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What is R o S?

Answer 1

$R\circ S =\{(x,z)~:~\exists y~\text{such that}~(x,y)\in S,(y,z)\in R\}$

$R=\{(\color{red}{a},b),(b,c),(\color{blue}{c},a)\}, S=\{(a,\color{blue}{c}),(c,\color{red}{a})\}$

So, as you found in the comments above, $R\circ S = \{(c,b),(a,a)\}$

Graphically, you may simply draw the set multiple times and arrows from one to the next. The composition is the set of paths to go from the furthest left to the furthest right, pictured in blue below.