91Ó°ÊÓ

Prove the following, where \(\mathrm{k}, \mathrm{m}, \mathrm{n}, \mathrm{q}\), and \(\mathrm{r}\) designate integers. Let \(\mathrm{n}>0\) and \(\mathrm{k}>0\). If \(\mathrm{q}\) is the quotient when \(\mathrm{m}\) is divided by \(\mathrm{n}\), and \(\mathrm{q}_{1}\) is the quotient when \(\mathrm{q}\) is divided by \(\mathrm{k}\), then \(\mathrm{q}_{1}\) is the quotient when \(\mathrm{m}\) is divided by \(\mathrm{nk}\).

In any ordered integral domain, define \(|a|\) by $$ |a|=\left\\{\begin{array}{rll} a & \text { if } & a \geq 0 \\ -a & \text { if } & a<0 \end{array}\right. $$ Using this definition, prove the following: $$ a \leq|a| $$

The purpose of this exercise is to give rigorous proofs (using induction) of the basic identities involved in the use of exponents or multiples. If \(A\) is a ring and \(a \in A\), we define \(\mathrm{n} \cdot a\) (where \(\mathrm{n}\) is any positive integer) by the pair of conditions: (i) \(1 \cdot a=a, \quad\) and (ii) \((\mathrm{n}+1) \cdot a=\mathrm{n} \cdot a+a\) Use mathematical induction (with the above definition) to prove that the following are true for all positive integers \(\mathrm{n}\) and all elements \(a, b \in A\) : $$ (\mathrm{n} \cdot a) b=a(\mathrm{n} \cdot b)=\mathrm{n} \cdot(a b) $$

In any ordered integral domain, define \(|a|\) by $$ |a|=\left\\{\begin{array}{rll} a & \text { if } & a \geq 0 \\ -a & \text { if } & a<0 \end{array}\right. $$ Using this definition, prove the following: $$ a \geq-|a| $$

Let \(A\) be an ordered integral domain. Prove the following, for all \(a, b\), and \(c\) in \(A\). $$ a^{2}+b^{2} \geq a b $$

In any ordered integral domain, define \(|a|\) by $$ |a|=\left\\{\begin{array}{rll} a & \text { if } & a \geq 0 \\ -a & \text { if } & a<0 \end{array}\right. $$ Using this definition, prove the following: $$ \text { If } b>0,|a| \leq b \text { iff }-b \leq a \leq b $$

The purpose of this exercise is to give rigorous proofs (using induction) of the basic identities involved in the use of exponents or multiples. If \(A\) is a ring and \(a \in A\), we define \(\mathrm{n} \cdot a\) (where \(\mathrm{n}\) is any positive integer) by the pair of conditions: (i) \(1 \cdot a=a, \quad\) and (ii) \((\mathrm{n}+1) \cdot a=\mathrm{n} \cdot a+a\) Use mathematical induction (with the above definition) to prove that the following are true for all positive integers \(\mathrm{n}\) and all elements \(a, b \in A\) : $$ \mathrm{m} \cdot(\mathrm{n} \cdot a)=(\mathrm{mn}) \cdot a $$

Let \(A\) be an ordered integral domain. Prove the following, for all \(a, b\), and \(c\) in \(A\). If \(a

$$ a^{2}+b^{2} \geq a b $$$$ a^{2}+b^{2} \geq-a b $$

Prove the following, where \(\mathrm{k}, \mathrm{m}, \mathrm{n}, \mathrm{q}\), and \(\mathrm{r}\) designate integers. In Theorem 3 , suppose \(m=n q_{1}+r_{1}=n q_{2}+r_{2}\) where \(0 \leq r_{1}, r_{2}