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

What is R o S?

Fitzwilliam
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  • Perhaps your book has a definition for composition of relations that you wouldn't mind sharing with us (to show us that you've at least tried to read the book before asking). The answer should become immediately apparent after reading the definition. – JMoravitz Jan 06 '16 at 21:42
  • since it is one set ... so i cant apply an arrow diagram as usual. i could figure out that RoS = {(c,b)} ?? i am not sure .. – Fitzwilliam Jan 06 '16 at 21:43
  • (c,b) is good, how did you get it? (because (c,a) is in $S$ and (a,b) is in $R$. Taking the left of a pair in $S$ and the right of a pair in $R$ whose other terms match). Are there any others? – JMoravitz Jan 06 '16 at 21:44
  • yessss it is (a,a) !!!! i got it ... – Fitzwilliam Jan 06 '16 at 21:48
  • if the question is S o R ?? i can figue it out {(b,a),(c,c)} . is that o.k.? – Fitzwilliam Jan 06 '16 at 21:51

1 Answers1

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

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

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.

enter image description here

JMoravitz
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