Question

State whether the following statements are TRUE or FALSE:
The union of two equivalence relations is also an equivalence relation.

• A. FALSE
• B. TRUE

Answer 1

The correct option is A FALSE

A relation is said to be equivalence relation is
(i) Reflexive
(ii) Symmetric and
(iii) Transitive
Union of two reflexive relations and two symmetric relations are reflexive and symmetric respectively. However, union of two transitive relations need not to be transitive. Therefore, union of two equivalence relations need not be an equivalence relation.
Example:
$Let{R}_{1}and{R}_{2}onsetA=\{1,2,3\}$
$R_1=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right\}$
is an equivalence relation
$R_2=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2\left)\right\}$
is not an equivalence relation.
$R_1\cup R_2=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2\left)\right\}$
is not an equivalence relation,
because (1, 2) & (2,3) needs (1, 3) element to be in transitive relation.