Write the equivalance relation on $A={1,2,3,4,5}$ induced by the partition $c ={{1,2},{3,4,5}}$.
Solution: ${(1,1)(1,2)(2,1)(2,2)(3,3)(3,4)(4,3)(4,4)(4,5)(5,4)(5,5)}$
Would this be the correct solution or did I miss something.
Equivalence relation induced by partition
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discrete-mathematics
3 Answers
1
You forgot $(3,5)$ and $(5,3)$
1
As you have (3,4) and (4,5) belongs to your relation. So you should have (3,5) in your relation for transitive.
As (3,5) so also (5,3) for symmetric.
0
Tip: The equivalence relation which produces such a partition, is the union of the Cartesian squares of the parts.
$$\begin{align} \bigcup_{p\in c}p^2 &= {\{1,2\}^2\cup\{3,4,5\}^2 }\\[1ex] & = { \{ {(1,1),(1,2),} \,{(2,1),(2,2),} \; {(3,3),(3,4),\color{red}{(3,5)},} \, {(4,3),(4,4),(4,5),} \, {\color{red}{(5,3)},(5,4),(5,5)}\}}\end{align}$$