14 Additional Exercises

Typeset the following.

1. $a<b$

2.

$$\left[\begin{array}{ccc}6& \frac{1}{2}& 0\\ 7& 5& 10\\ 0& 17& 0\end{array}\right]$$

3. $\left\{\frac{ab}{c}\right\}$

4. $\{\frac{ab}{c}\}$

5.

$\begin{array}{}\text{(1)}& a_{11}+b_{11}& =c_{11}& a_{12}& =b_{12}\text{(2)}& b_{21}& =c_{21}& a_{22}& =b_{22}+c_{22}\end{array}$

6.

$\begin{array}{}\text{(3)}& a& =b+c-d\text{(4)}& & +e-f\text{(5)}& & =g+h\text{(6)}& & =i\end{array}$

7. $$\begin{array}{}\text{(7)}& x& =y& X& =Y& a& =b+c\text{(8)}& x^{\prime }& =y^{\prime }& X^{\prime }& =Y^{\prime }& a^{\prime }& =b\text{(9)}& x+x^{\prime }& =y+y^{\prime }& X+X^{\prime }& =Y+Y^{\prime }& a^{\prime }b& =c^{\prime }b\end{array}$$

8. $$\begin{array}{}\text{(10)}& x& =y_1-y_2+y_3-y_5+y_8-\dots & & \text{by 6.6}\text{(11)}& & =y^{\prime }\circ y^{\ast }& & \text{by 7.5}\text{(12)}& & =y(0)y^{\prime }& & \text{by Axiom 1.}\end{array}$$

9. $$\begin{array}{}\text{(13)}& A_1& =N_0(\lambda ;{\mathrm{\Omega }}^{\prime })-\varphi (\lambda ;{\mathrm{\Omega }}^{\prime }),\text{(14)}& A_2& =\varphi (\lambda ;{\mathrm{\Omega }}^{\prime })-\varphi (\lambda ;\mathrm{\Omega }),\text{(15)}& A_3& =\mathcal{N}(\lambda ;\omega ).\end{array}$$

10.

$\left(\begin{array}{cc}2\tau & 7\varphi -\frac{5}{12}\\ 3\psi & \frac{\pi }{8}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\text{and}\left[\begin{array}{ccr}3& 4& 5\\ 1& 3& 729\end{array}\right]$

11. Equation $\sum _{i=0}^{\mathrm{\infty }}{a}_i{x}^i$ is a typical power series.