Abstract Algebra Theory and Applications - Computer Science ...

18 CHAPTER 0 PRELIMINARIESIf we consider the equivalence relation established by the integers modulo3, then[0] = {. . . , −3, 0, 3, 6, . . .},[1] = {. . . , −2, 1, 4, 7, . . .},[2] = {. . . , −1, 2, 5, 8, . . .}.Notice that [0] ∪ [1] ∪ [2] = Z and also that the sets are disjoint. The sets[0], [1], and [2] form a partition of the integers.The integers modulo n are a very important example in the study ofabstract algebra and will become quite useful in our investigation of variousalgebraic structures such as groups and rings. In our discussion of theintegers modulo n we have actually assumed a result known as the divisionalgorithm, which will be stated and proved in Chapter 1.Exercises1. Suppose thatA = {x : x ∈ N and x is even},B = {x : x ∈ N and x is prime},C = {x : x ∈ N and x is a multiple of 5}.Describe each of the following sets.(a) A ∩ B(b) B ∩ C(c) A ∪ B(d) A ∩ (B ∪ C)2. If A = {a, b, c}, B = {1, 2, 3}, C = {x}, and D = ∅, list all of the elements ineach of the following sets.(a) A × B(b) B × A(c) A × B × C(d) A × D3. Find an example of two nonempty sets A and B for which A × B = B × Ais true.4. Prove A ∪ ∅ = A and A ∩ ∅ = ∅.5. Prove A ∪ B = B ∪ A and A ∩ B = B ∩ A.6. Prove A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).