1 Show, ifnNis not prime, then there exists[a],[b]Zsuch that[a][0][b], but[a][b]=[0]. Show, ifpNis prime,[a],[b]Zand[a][0][b], then[a][b][0].

1

• Show, if n∈N is not prime, then there exists [a],[b]∈Z such that [a]≠[0]≠[b], but [a]⋅[b]=[0].
• Show, if p∈N is prime, [a],[b]∈Z and [a]≠[0]≠[b], then [a]⋅[b]≠[0].

2. Suppose R and S are two equivalence relations on a set A. Prove that R ∩ S is also an equivalence relation. (For an example of this, look at Figure 11.3. Observe that for the equivalence relations R2,R3 and R4, we have R2 ∩ R3 = R4.)

3. Is the function θ : P(Z) → P(Z) defined as θ(X) = X bijective? If so, find θ −1 .

4. Given f : A → B and subsets Y,Z ⊆ B, prove f −1 (Y ∩ Z) = f −1 (Y)∩ f −1 (Z)