If a, b are elements of a ring and m, \(n \in Z\), show that \((na) (mb) = (mn) (ab)\)

Explain why (a) \(Z_9\) is not isomorphic to \(Z_3 \times Z_3\); (b) \(Z_9 \times Z_9\) is not isomorphic to \(Z_9 \times Z_3 \times Z_3\).

In Exercises 6 through 9, write the set in the form \(\{x|P(x)\}\), where P(x) is a property that describes the elements of the set. 6. \(\{2, 4, 6, 8, 10\}\)

7. \(\{a, e, i, o, u\}\) 8. \(\{1, 8, 27, 64, 125\}\) 9. \(\{-2, -1, 0, 1, 2\}\)

Question 4 (Module Outcome #4): Find the best-case, worst-case and average-case number of < comparisons are performed by the following piece of pseudocode. Precondition: \(n\in\{1,3,5,7,9\}\ \text{while}\ n < 6\ \text{do}\ n\leftarrow n+3\)

In the froup \(\displaystyle{Z}_{{12}}\), find \(|a|, |b|\), and \(|a+b|\) \(a=5, b=4\)

Prove the following. (1) \(Z \times 5\) is a cyclic group. (2) \(Z \times 8\) is not a cyclic group.

Let H be a normal subgroup of a group G, and let \(m = (G : H)\). Show that \(a^{m} \in H\) for every \(a \in G\)

If U is a set, let \(\displaystyle{G}={\left\lbrace{X}{\mid}{X}\subseteq{U}\right\rbrace}\). Show that G is an abelian group under the operation \(\oplus\) defined by \(\displaystyle{X}\oplus{Y}={\left({\frac{{{x}}}{{{y}}}}\right)}\cup{\left({\frac{{{y}}}{{{x}}}}\right)}\)

Show that the prime subfield of a field of characteristic p is ringisomorphic to \(Z_{p}\) and that the prime subfield of a field of characteristic 0 is ring-isomorphic to Q.