# Sets, Relations and Function - Equivalence relation

Let A ={1,2,3,4,5,6,7} which of the following two is a partition giving rise to an equivalence relation ? Why ?

(a) $A_1 = \{1,3,5\}; A_2 = \{2\} ; A_3 = \{4,7\}$

(b) $B_1 =\{1,2,5,7\} ; B_2 =\{3\} ; B_3 =\{4,6\}$

Since for an equivalence relation : a relation is said to be equivalence if it possess three properties : reflexive, symmetric, and transitive.

How to use this here... please guide

• Do you know what "partition" means? – Chris Eagle Mar 11 '13 at 16:49

## 2 Answers

(a) $A_1 = \{1,3,5\}; A_2 = \{2\} ; A_3 = \{4,7\}$ since $$A_1\cup A_2\cup A_3\neq\{1,2,3,4,5,6,7\}=A$$ $$\{A_1,A_2,A_3\}$$ is not a partition of $A$

(b) $B_1 =\{1,2,5,7\} ; B_2 =\{3\} ; B_3 =\{4,6\}$

Because $B_1\cap B_2\cap B_3=\emptyset$ and $B_1\cup B_2\cup B_3=A$ then

$$\{B_1, B_2, B_3\}$$ is a partition of $A$

Hint: for a partition, each element of the set must appear in exactly one subset. Which one satisfies this? What requirement for an equivalence relation would fail in the other case?