QUESTION $1[40]$ Assignment $2$

QUESTION $1$

$1\mathrm{.}1$ Convert Binary $1100\mathrm{.}11$ to Decimal.

$1\mathrm{.}2$ Convert $7538$ to Hexadecimal.

$1\mathrm{.}3$ Using Boolean algebra techniques, simplify the following expression.

$\frac{?}{\text{b}\text{a}\text{r}}(AB)(\frac{?}{\text{b}\text{a}\text{r}}(A)+B)(\frac{?}{\text{b}\text{a}\text{r}}(B)+B)+AB+\frac{B}{b}ar(A)$

$1\mathrm{.}4$ Determine the $2\text{'}$s complement of ${1111001}_2$

$1\mathrm{.}5$ Perform the following additions:

$($a$){247}_8+{735}_8$

$($b$){1111}_2+{1101}_2$

$($c$)8A{B}_{16}+B{78}_{16}$

$1\mathrm{.}6$ Write the expression in sigma notation:

$x_{(ABCD)}=A\frac{B}{b}ar\frac{C}{b}ar(D)+\frac{A}{b}ar(B)CD\frac{+}{b}ar(A)B\frac{C}{b}ar(D)\frac{+}{b}ar(AB)\frac{C}{b}ar(D)+\frac{A}{b}ar\frac{B}{b}ar\frac{C}{b}ar(D)$

$1\mathrm{.}7$ What is the output of a NAND gate when two inputs are HIGH?

QUESTION $2$ QUESTION $2$

$2\mathrm{.}1$ Simplify the expression

$\frac{?}{\text{b}\text{a}\text{r}}(W)\frac{x}{b}ar\frac{Y}{b}ar(Z)+W\frac{x}{b}ar(Y)Z+W\frac{x}{b}ar\frac{Y}{b}ar(Z)+W\frac{\stackrel{}{x}}{b}ar(Y)Z\frac{+}{b}ar(W)\stackrel{}{x}YZ\frac{+}{b}ar(W)xYZ+WxYZ+Wx\frac{Y}{b}ar(Z)$

using a Karnaugh map.

Please note: use WX in the rows and $YZ$ in the columns

$2\mathrm{.}2$ Write the expanded forms of the terms in the given expression:

$F(ABCD)=\frac{A}{b}ar(C)+$ABC $D$

$2\mathrm{.}3$ Implement the logic circuit for the expression

$F(WxY)=(\frac{?}{\text{b}\text{a}\text{r}}(W)+\stackrel{}{x}+Y)(\frac{?}{\text{b}\text{a}\text{r}}(W)\stackrel{}{x}+Y)\frac{+}{b}ar(W)xY.$ Please note: Do not simplify the expression.

$2\mathrm{.}4$ Draw the truth table of a $2-$input NOR gate.

Question $3.2\mathrm{.}4$ Draw the truth table of a $2-$input NOR gate.

Question $3.$

$3\mathrm{.}1($a$)$ Convert the given Boolean expression to standard SOP $($Sum of Products$)$ form.

$x(ABC)=\frac{A}{b}ar(B)C\frac{+}{b}ar\frac{A}{b}ar(C)+$ABCD$+BCD$

$($b$)$ Write the Boolean expression for the following Logic circuit:

$3\mathrm{.}2($a$)$ Discuss the Positive edge triggered J$-$K flip flop and also provide its logic circuit.

$($b$)$ Draw the truth table of the Positive edge triggered J$-$K flip flop.

$3\mathrm{.}3$ Answer the following two questions:

$($a$)$ Apply Demorgan's theorems to the expression: $x\frac{?}{b}=ar(((\frac{?}{\text{b}\text{a}\text{r}}(A))+B)(\frac{?}{\text{b}\text{a}\text{r}}(AC)))$

$($b$)$ Prove that $\frac{?}{\text{b}\text{a}\text{r}}((A+BC))+(\frac{?}{\text{b}\text{a}\text{r}}(\frac{?}{\text{b}\text{a}\text{r}}(AB)))=\frac{A}{b}ar(B)$

$1\mathrm{.}1$ Convert Binary $1100\mathrm{.}11$ to Decimal. $(6)$

$1\mathrm{.}2$ Convert $7538$ to Hexadecimal. $(6)$

$1\mathrm{.}3$ Using Boolean algebra techniques, simplify the following expression. $(6)$

AB $($A

$$

$+$B$)($B

$$

$+$B$)+$AB$+$BA$$

$1\mathrm{.}4$ Determine the $2$s complement of $1111001(3)$

$1\mathrm{.}5$ Perform the following additions:

$($a$)2478+7358(4)$

$($b$)1111+1101(4)$

$($c$)8$AB$16+$ B$7816(4)$

$1\mathrm{.}6$ Write the expression in sigma notation:

X$()=$ ABC$$

D$$

$+$ AB$$

CD $+$A

$$

BCD$$

$+$A

$$

B$$

CD$$

$+$ AB$$

C$$

D$$

$(5)$

$1\mathrm{.}7$ What is the output of a NAND gate when two inputs are HIGH? $(2)$

QUESTION $2[30]$

$2\mathrm{.}1$ Simplify the expression

W$$

X Y

$$

Z$$

$+$ W X Y

$$

Z $+$ W X Y

$$

Z$$

$+$ W X

$$

Y$$

Z $+$ W

$$

X$$

Y Z $+$ W

$$

X Y Z $+$ W X Y Z $+$ W X Y Z

$$

using a Karnaugh map. $(15)$

Please note: use WX in the rows and YZ in the columns

$2\mathrm{.}2$ Write the expanded forms of the terms in the given expression: $(5)$

F $($ABCD$)=$ AC$$

$+$A B C D

$2\mathrm{.}3$ Implement the logic circuit for the expression

F$($WXY$)=($W

$$

$+$X

$$

$+$Y$)($W X$+$ Y $)+$ W

$$

X Y$.$ Please note: Do not simplify the expression. $(8)$

$2$ ICT$2632$

$2\mathrm{.}4$ Draw the truth table of a $2-$input NOR gate. $(2)$

Question $3.[30]$

$3\mathrm{.}1($a$)$ Convert the given Boolean expression to standard SOP $($Sum of Products$)$ form. $(8)$

X$($ABC$)=$AB$$

C$+$A $$C

$$

$+$ABCD$+$ BCD

$($b$)$ Write the Boolean expression for the following Logic circuit: $(2)$

$3\mathrm{.}2($a$)$ Discuss the Positive edge triggered J$-$K flip flop and also provide its logic circuit. $(8)$

$($b$)$ Draw the truth table of the Positive edge triggered J$-$K flip flop. $(4)$

$3\mathrm{.}3$ Answer the following two questions:

$($a$)$ Apply Demorgan's theorems to the expression: X $=($A

$$

$+$B$)($A$$C$)(4)$

$($b$)$ Prove that $($A $+$ BC$)+($AB$$

$)=$ AB$$

$(4)$

TOTAL $[100$ Marks$]$