(a) Consider the group (Z2 × Z2, ⊕) where, for a, b, c, d ∈ Z2, (a, b) ⊕ (c, d) = (a + c, b + d) - the sums a + c and b + d are computed using addition modulo 2. What is the value of (1, 0) ⊕ (0, 1) ⊕ (1, 1) in this group?
(b) Now consider the group (Z2 × Z2 × Z2, ⊕) where (a, b, c) ⊕ (d, e, f) = (a + d, b + e, c + f). (Here the sums a + d, b + e, c + f are computed using addition modulo 2.) What do we obtain when we add the seven nonzero (or nonidentity) elements of this group? (c) State and prove a generalization that includes the results in parts (a) and (b).