I am having trouble with these compositions. $$T = \{(a,a), (a,b), (b,c), (b,d), (c,d), (d,a), (d,b)\}$$ $$U = \{(a,a), (a,d), (b,c), (b,d), (c,a), (d,d)\}$$

I need to find $T \circ T$, $U \circ T$, and $T \circ U$.

My problem is when I get down to, for example $U \circ T$ where $(d,a)$ corresponds with both $(a,a)$ and $(a,c)$. This seems to happen for everyone of these problems. Is it even possible to take the composition of these?

  • $\begingroup$ What is your definition of composition with set elements? What are the functions? $\endgroup$ – karakfa Feb 8 '16 at 20:01
  • $\begingroup$ I am not sure what you mean by "$(d,a)$ corresponds with both $(a,a)$ and $(a,c)$". $~$ What are you trying to do? $\endgroup$ – Graham Kemp Feb 8 '16 at 20:39

Yes, it is possible to find these compositions.

$$\begin{align} U\circ T ~&\mathop{:=}~ \{(x,z): \exists y~((x,y)\in T \,\wedge\, (y,z)\in U)\} \\T &= \{(a,a), (a,b), (b,c), (b,d), (c,d), (d,a), (d,b)\} \\U &= \{(a,a), (a,d), (b,c), (b,d), (c,a), (d,d)\} \\U\circ T &= \{(a,a), (a,c), (a,d), \ldots\textsf{et cetera}\} \end{align}$$

For instance $(a,c)\in U\circ T$ because $(a,b)\in T$ and $(b,c)\in U$.   As long as there is at least one such transition, the element appears in the composition.

Can you find the rest?

  • $\begingroup$ Yes, thank you. $\endgroup$ – Jake Mager Feb 8 '16 at 20:44

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