Question $3$

$($a$)$ Given a Boolean expression, $x=xYZ+\frac{x}{b}ar(Y)(\frac{?}{\text{b}\text{a}\text{r}}\frac{\stackrel{}{x}}{b}ar(\frac{?}{\text{b}\text{a}\text{r}}(Z))).$

$($i$)$ Simplify the expression using Boolean Law.

$(7$ marks$)$

$($ ii $)$ Construct a combinatorial logic circuit using inverters, OR gates, and AND gates for the expression $x.$

$(5$ marks$)$

$($ iii$)$ Give two applications of Boolean algebra in Information Technology.

$(2$ marks$)$

$($b$)($i$)$ Find the output of the combinatorial circuits figure below:

$(1$ mark$)$

$($ ii $)$ Use a K$-$map to minimize the following sum$-$of$-$products expression:

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

$x=\frac{x}{b}ar(y)z+x\frac{y}{b}ar(z)+\stackrel{}{x}yz+\stackrel{}{x}\frac{y}{b}ar(z)+\frac{\stackrel{}{x}}{b}ar\frac{z}{b}ar(z)$

$(9$ marks$)$

$5$Question $4$

$($a$)$ Determine whether the following sentences are propositions or not.

$($i$)$ Ts Dr Soong conducts the test for students on Weekend.

$(1$ mark$)$

$($ ii $)$ Please ensure that you carefully read the question and record your answer on

the answer sheets.

$(1$ mark$)$

$($ iii $)m+n=p$

$(1$ mark$)$

$($iv$)109+361=400$

$(1$ mark$)$

$($v$)$ Please ensure that you study diligently for MAT$2204.$

$(1$ mark$)$

$($b$)$ Let $p,q$ and $r$ be the propositions. Write each of the following statements in logical

notation form.

p: Ming Hooi obtain an $A+$ in the final exam.

$q$ : Ming Hooi does each exercise in this book.

$r$ : Ming Hooi obtain an A $+$ in Ts Dr Soong' class.

$($i$)$ Ming Hooi didn't get an A in this class.

$(1$ mark$)$

$($ ii $)$ Do every exercise in this book is necessary and sufficient for Ming Hooi get an

A in the final exam.

$(2$ marks$)$

$($ iii $)$ If Ming Hooi do every exercise in this book and get an A in the final exam, then

she get an A in Ts Dr Soong' class.

$(2$ marks$)$

$($c$)$ Write the truth table for the proposition:

$($i$)$ not$(notpq)(notq)$

$(7$ mark$)$

$($ ii $),(pq)(notpq)$

$(7$ marks$)$

$($ iii $)$ Hence, determine whether the two truth tables are equivalence.