Let a set A = A_1 A_2 A_k, where A_i A_j= for i j, 1 i,j k.

Question

Let a set $A = A_1\cup A_2\cup\cdots\cup A_k$, where $A_i\cap A_j=\emptyset$ for $i
eq j$, $1\leq i,j\leq k$. Define the relation $R$ from $A$ to $A$ by $R = \{(x,y) : y\in A_i 	ext{ if and only if } x\in A_i,\ 1\leq i\leq k\}$. Then, $R$ is:

Accepted Answer

- Reflexive: $(a,a)$: $a\in A_i \iff a\in A_i$, trivially true.
- Symmetric: if $(a,b)\in R$, i.e. $a\in A_i \iff b\in A_i$, the same biconditional gives $(b,a)\in R$.
- Transitive: if $(a,b)\in R$ and $(b,c)\in R$, then $a\in A_i\iff b\in A_i$ and $b\in A_i \iff c\in A_i$, so $a\in A_i \iff c\in A_i$, giving $(a,c)\in R$.

$R$ is reflexive, symmetric, and transitive --- an equivalence relation.

Solutions · 1