COMP 2611 Homework 1

Q1

a

$$\begin{array}{rl}AB(\bar{A}\bar{E}+\bar{E}\bar{B})+ABCD(\overline{C+D+E})& =AB\bar{A}\bar{E}+AB\bar{E}\bar{B}+ABCD(\bar{C}\bar{D}\bar{E})\\ & =A\bar{A}B\bar{E}+AB\bar{B}\bar{E}+ABC\bar{C}D\bar{D}\bar{E}\\ & =0+0+0\\ & =0\end{array}$$

b

$$\begin{array}{rl}(A+C)(\bar{A}+B)(C+B)+A(\bar{A}+B)& =(\bar{A}+B)((A+C)(C+B)+A)\\ & =(\bar{A}+B)(AC+AB+CC+BC+A)\\ & =(\bar{A}+B)(A+C+AB+AC+BC)\\ & =(\bar{A}+B)(A+AB+C+CA+BC)\\ & =(\bar{A}+B)(A+C+BC)\\ & =\bar{A}A+\bar{A}C+\bar{A}BC+AB+BC+BBC\\ & =0+\bar{A}C+\bar{A}CB+BC+BC+AB\\ & =\bar{A}C+BC+AB\\ & =\bar{A}C+AB+BC\\ & =\bar{A}C+AB+(A+\bar{A})BC\\ & =AB+ABC+\bar{A}BC+\bar{A}C\\ & =AB(1+C)+\bar{A}C(1+B)\\ & =AB+\bar{A}C\end{array}$$

c

$$\begin{array}{rl}& (A\bar{B}(D+BD)+\bar{A}\bar{B})D+(C\bar{D}+B\bar{D})\overline{(\bar{D}+\bar{D})}\\ & =(A\bar{B}D+A\bar{B}BD+\bar{A}\bar{B})D+(C\bar{D}+B\bar{D})D\\ & =(A\bar{B}D+\bar{A}\bar{B})D+(C\bar{D}D+B\bar{D}D)\\ & =A\bar{B}D+\bar{A}\bar{B}D\\ & =\bar{B}D(A+\bar{A})\\ & =\bar{B}D\end{array}$$

Q2

a

$x_1$	$x_0$	$y_1$	$y_0$	Minterm	Maxterm	$f(X)$	$f(Y)$	$Z$
0	0	0	0	$m_0$	$M_0$	1	1	1
0	0	0	1	$m_1$	$M_1$	1	2	1
0	0	1	0	$m_2$	$M_2$	1	3	1
0	0	1	1	$m_3$	$M_3$	1	4	1
0	1	0	0	$m_4$	$M_4$	2	1	1
0	1	0	1	$m_5$	$M_5$	2	2	0
0	1	1	0	$m_6$	$M_6$	2	3	1
0	1	1	1	$m_7$	$M_7$	2	4	0
1	0	0	0	$m_8$	$M_8$	3	1	1
1	0	0	1	$m_9$	$M_9$	3	2	1
1	0	1	0	$m_{10}$	$M_{10}$	3	3	0
1	0	1	1	$m_{11}$	$M_{11}$	3	4	1
1	1	0	0	$m_{12}$	$M_{12}$	4	1	1
1	1	0	1	$m_{13}$	$M_{13}$	4	2	0
1	1	1	0	$m_{14}$	$M_{14}$	4	3	1
1	1	1	1	$m_{15}$	$M_{15}$	4	4	0

b

• Sum-of-Product (SoP)

$$Z=m_0+m_1+m_2+m_3+m_4+m_6+m_8+m_9+m_{11}+m_{12}+m_{14}$$

• Product-of-Sum (PoS)

$$Z=M_5{M}_7{M}_{10}{M}_{13}{M}_{15}$$

c

$$Z=\overline{x_1}\overline{x_0}+\overline{y_1}\overline{y_0}+x_0\overline{y_0}+\overline{x_0}{y}_0$$

$$Z=\overline{x_1}\overline{x_0}+x_0\overline{y_0}+\overline{x_0}\overline{y_1}+\overline{x_0}{y}_0$$

d

Q3

$A$	$B$	$Q_t$	$Q_{t+1}$
0	0	0	Latch
0	0	1	Latch
0	1	0	Latch
0	1	1	Latch
1	0	0	Latch
1	0	1	0
1	1	0	1
1	1	1	Latch

Q4