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Suppose $A = \{a, b, c, d\}$ and $\Pi_1$ is the following partition of A

$\Pi_1 = \left\{\left\{a, b, c\right\}\left\{d\right\}\right\}$

  1. List the ordered pairs of the equivalence relations induced by $\Pi_1$.

  2. Draw the graph of the above equivalence relation.

  3. Let $\Pi_2 = \left\{\left\{a\right\}, \left\{b\right\}, \left\{C\right\}, \left\{d\right\}\right\}$

     $\Pi_3 = \left\{\left\{a, b, c, d\right\}\right\}$

    and $\Pi_4 = \left\{\left\{a, b\right\}, \left\{c,d\right\}\right\}$

    Draw a Poset diagram of the poset, $\left\langle\left\{\Pi_1, \Pi_2, \Pi_3, \Pi_4\right\}, \text{ refines } \right\rangle$.

 

asked in Set Theory & Algebra by Veteran (66.2k points) 1154 2199 2522
edited by | 441 views

2 Answers

+3 votes
(a) the ordered pairs of the equivalence relations induced = { (a,a) (a,b) (a,c) (b,a) (b,b) (b,c) (c,a) (c,b) (c,c)     (d,d) }

 

ps : equivalence relations = each partition power set - phi
answered by Active (2.2k points) 2 10 31
we can find the same by { {a,b,c} ${\times }$ {a,b,c} , {d} ${\times }$  {d} }

= { (a,a), (a,b), (a,c), (b,a), (b,b), (b,c), (c,a), (c,b) ,(c,c), (d,d) }
what about (b) and (c) ?
+2 votes
answered by Active (1.5k points) 1 8 22


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