Chapter 21

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 \(a, b \in A\) : $$ \mathrm{n} \cdot(a+b)=\mathrm{n} \cdot a+\mathrm{n} \cdot b $$

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

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: $$ |-a|=|a| $$

Let \(A\) be an integral system. Let \(h: \mathbb{Z} \rightarrow A\) be defined by: \(h(\mathrm{n})=\mathrm{n} \cdot 1\). The purpose of this exercise is to prove that \(h\) is an isomorphism, from which it follows that \(A \cong \mathbb{Z} .\) Prove the following: For every positive integer \(\mathrm{n}, \mathrm{n} \cdot 1>0 .\) From this, deduce that \(A\) has nonzero characteristic.

Prove each of the following, using the principle of mathematical induction $$ 1+3+5+\cdots+(2 n-1)=n^{2} $$ (The sum of the first \(\mathrm{n}\) odd integers is \(\mathrm{n}^{2}\).)

Let \(A\) be an ordered integral domain. Prove the following, for all \(a, b\), and \(c\) in \(A\). $$ a^{2}-2 a b+b^{2} \geq 0 $$

Prove the following in \(\mathbb{Z}\) : Let \(K\) denote a set of positive integers. Consider the following conditions: (i) \(1 \in K\). (ii) For any positive integer \(\mathrm{k}\), if every positive integer less than \(\mathrm{k}\) is in \(K\), then \(\mathrm{k} \in K\). If \(K\) satisfies these two conditions, prove that \(K\) contains all the positive integers.

Prove the following in \(\mathbb{Z}\) : Let \(S_{\mathrm{n}}\) represent any statement about the positive integer \(\mathrm{n} .\) Consider the following conditions: (i) \(S_{1}\) is true. (ii) For any positive integer \(\mathrm{k}\), if \(S_{\mathrm{i}}\) is true for every \(\mathrm{i}<\mathrm{k}, S_{\mathrm{k}}\) is true. If Conditions (i) and (ii) are satisfied, prove that \(S_{n}\) is true for every positive integer \(\mathrm{n}\).

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 \(a, b \in A\) : $$ (\mathrm{n}+\mathrm{m}) \cdot a=\mathrm{n} \cdot a+\mathrm{m} \cdot a $$

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: $$ a \leq|a| $$

Let \(A\) be an ordered integral domain. Prove the following, for all \(a, b\), and \(c\) in \(A\). $$ 1^{3}+2^{3}+\cdots+n^{3}=(1+2+\cdots+n)^{2} $$

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 \(A\). $$ a^{2}+b^{2} \geq a b $$

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 \(a, b \in A\) : $$ \mathrm{m} \cdot(\mathrm{n} \cdot a)=(\mathrm{mn}) \cdot a $$

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}

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: $$ \text { If } b>0,|a| \leq b \text { iff }-b \leq a \leq b $$

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

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

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: $$ |a+b| \leq|a|+|b| $$

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

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 \(a, b \in A\) : $$ (\mathrm{n} \cdot a)(\mathrm{m} \cdot b)=(\mathrm{nm}) \cdot a b \quad \text { (Use parts } 3 \text { and 4.) } $$

Prove the following, where \(\mathrm{k}, \mathrm{m}, \mathrm{n}, \mathrm{q}\), and \(\mathrm{r}\) designate integers. If \(\mathrm{r}\) is the remainder when \(\mathrm{m}\) is divided by \(\mathrm{n}\), then \(\overline{\mathrm{m}}=\overline{\mathrm{r}}\) in \(\mathbb{Z}_{\mathrm{n}} ;\) and conversely.

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: $$ |a-b| \leq|a|+|b| $$

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

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: $$ |a b|=|a| \cdot|b| $$

Let \(A\) be an ordered integral domain. Prove the following, for all \(a, b\), and \(c\) in \(A\). $$ a+b \leq a b+1, \text { if } a, b \geq 1 $$

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: $$ |a|-|b| \leq|a-b| $$

Let \(A\) be an ordered integral domain. Prove the following, for all \(a, b\), and \(c\) in \(A\). The Fibonacci sequence is the sequence of integers \(F_{1}, F_{2}, F_{3}, \ldots\) defined as follows: \(F_{1}=1 ; F_{2}=1 ; F_{\mathrm{n}+2}=F_{n+1}+F_{\mathrm{n}}\) for all positive integers \(\mathrm{n}\). (That is, every number, after the second one, is the sum of the two preceding ones.) Use induction to prove that for all \(\mathrm{n}>0\) $$ F_{\mathrm{n}+1} F_{\mathrm{n}+2}-F_{\mathrm{n}} F_{\mathrm{n}+3}=(-1)^{\mathrm{n}} $$

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: $$ || a|-| b|| \leq|a-b| $$