Final answer:
We proved that R_1 \oplus R_2 is irreflexive by using the definition of reflexive relations and the property of symmetric difference. Since R_1 and R_2 are reflexive, all pairs (a, a) are in both R_1 and R_2, and these pairs are therefore not included in R_1 \oplus R_2, making it irreflexive.
Step-by-step explanation:
To show that R_1 \oplus R_2 is irreflexive given that R_1 and R_2 are reflexive relations on a set A, we first need to understand the definition of reflexive and irreflexive relations, along with the operation symbolized by \oplus, which in this context denotes the symmetric difference of two relations.
A relation R on a set A is reflexive if every element a in A is related to itself, that is, (a, a) is in R for all a in A. An irreflexive relation is one where no element is related to itself, so (a, a) is not in the relation for any a in A.
The symmetric difference of two relations R_1 and R_2, denoted R_1 \oplus R_2, contains all the pairs that are in either R_1 or R_2 but not in both. Since R_1 and R_2 are reflexive, every element (a, a) is in both R_1 and R_2. Therefore, these pairs will not be in the symmetric difference, making R_1 \oplus R_2 irreflexive.