Question

To determine if the Boolean sum of two equivalence relations is an equivalence relation.

Short answer

In general Boolean sum of two equivalence relations is not an equivalence relation.

Step-by-step solution

Given data

\(R\) and \(T\) are the equivalence relations on a set \(S\).

Formula used of Boolean sum

In Boolean Algebra, the addition of two values is equivalent to the logical OR function there by producing a "Sum" term when two or more input variables or constants are "OR'ed" together. In other words, in Boolean Algebra the OR function is the equivalent of addition and so its output state represents the "Sum" of its inputs.

Find the equivalence relation

Let \(R\) and \(T\) be the equivalence relations on a set \(S\).

As \(R\) and \(T\) are both reflexive,

\((x,x) \in R,(x,x) \in T \Rightarrow (x,x) \notin R \oplus T\)

Thus, \(R \oplus T\) is never reflexive, which implies that it is not an equivalence relation.

Boolean sum of two equivalence relation is not an equivalence relation.