Chapter 4, Problem 2P

Program Plan Intro

Logic gate:

• Logic gate is an electronic circuit that is used to take logical decisions based on the input.
• It contains one or more number of inputs and one output.
• The working of logic gate is based on the binary principle that has two states either logic 0 or logic 1.
• The output of logic gate is produced when it satisfies any of its logic conditions.
• The logic condition depends upon the type of the gates and the number of inputs.
• The primary logic gates include AND, OR and NOT and the combinations of these gates are used to implement any of the other logic gates.

AND gate:

• The AND gate refers to a logic gate whose output will be HIGH only when all the inputs are HIGH.
• The output of AND gate will be LOW when any one of its input is LOW.
• The symbol to represent AND gate is given below.

• The truth table for AND gate is as follows.

INPUT A	INPUT B	OUTPUT Y
0	0	0
0	1	0
1	0	0
1	1	1

OR gate:

• The OR gate refers to a logic gate whose output will be HIGH when any one of its inputs are HIGH.
• The output of AND gate will be LOW when both the inputs are LOW.
• The symbol to represent OR gate is given below.

• The truth table for OR gate is as follows.

INPUT A	INPUT B	OUTPUT Y
0	0	0
0	1	1
1	0	1
1	1	1

NOT gate:

• The NOT gate refers to a logic gate whose output will be HIGH when it’s input is LOW and whose output will be LOW when it’s input is HIGH.
• The symbol to represent NOT gate is given below.

• The truth table for NOT gate is as follows.

INPUT A	OUTPUT Y
0	1
1	0

Explanation of Solution

a.

Logic gate circuit:

The logic gate circuit is as follows.

Explanation:

In the above given logic gate circuit,

• The inputs “A” and “B” are connected to logic OR gate and the corresponding output will be (A+B).
• Now, the resultant along with other inputs “C” and “D” are connected to logic AND gate whose output will be $Y=(A+B)CD$.

Therefore, the Boolean expression for the given logic circuit is $Y=(A+B)CD$.

Explanation of Solution

b.

Logic gate circuit:

The logic gate circuit is as follows.

Explanation:

In the above given logic gate circuit,

• The inputs “A” and “B” are connected to logic OR gate and the corresponding output will be (A+B).
• The input “C” is connected to logic NOT gate and the corresponding output will be “$\overline{C}$”.
• Then, the inputs “$\overline{C}$” and “D” are connected to another logic OR gate and the corresponding output will be $(\overline{C}+D)$.
• Now, the output (A+B) and $(\overline{C}+D)$ are connected to a logic AND gate whose output will be $Y=(A+B)(\overline{C}+D)$.

Therefore, the Boolean expression for the given logic circuit is $Y=(A+B)(\overline{C}+D)$.

Explanation of Solution

c.

Logic gate circuit:

The logic gate circuit is as follows.

Explanation:

In the above given logic gate circuit,

• The inputs “A” and “B” are connected to logic AND gate and the corresponding output will be (AB).
• Now, the resultant along with other input “C” is connected to logic OR gate whose output will be $Y=(AB+C)$.

Therefore, the Boolean expression for the given logic circuit is $Y=(AB+C)$.

Explanation of Solution

d.

Logic gate circuit:

The logic gate circuit is as follows.

Explanation:

In the above given logic gate circuit,

• The inputs “A” and “B” are connected to logic AND gate and the corresponding output will be (AB).
• Now, the resultant along with other input “C” is connected to logic OR gate whose output will be $(AB+C)$.
• The input “E” is connected to logic NOT gate and the corresponding output will be “$\overline{E}$”.
• Now, the outputs $(AB+C)$ and $\overline{E}$ along with the input “D” are connected to a logic AND gate whose output will be $Y=[(AB+C)D\overline{E}]$.

Therefore, the Boolean expression for the given logic circuit is $Y=[(AB+C)D\overline{E}]$.

Explanation of Solution

e.

Logic gate circuit:

The logic gate circuit is as follows.

Explanation:

In the above given logic gate circuit,

• The inputs “A” and “B” are connected to logic OR gate and the corresponding output will be (A+B).
• Then, the inputs “C” and “D” are connected to logic OR gate and the corresponding output will be $(C+D)$.
• Now, the output (A+B) and $(C+D)$ are connected to a logic AND gate whose output will be $Y=(A+B)(C+D)$.

Therefore, the Boolean expression for the given logic circuit is $Y=(A+B)(C+D)$.

Explanation of Solution

f.

Logic gate circuit:

The logic gate circuit is as follows.

Explanation:

In the above given logic gate circuit,

• The input “B” is connected to logic NOT gate and the corresponding output will be “$\overline{B}$”.
• The inputs “A” and “$\overline{B}$” are connected to logic AND gate and the corresponding output will be ($A\overline{B}$).
• Similarly, the input “D” is connected to logic NOT gate and the corresponding output will be “$\overline{D}$”.
• The inputs “C” and “$\overline{D}$” are connected to logic AND gate and the corresponding output will be ($C\overline{D}$).
• Then, the input “F” is connected to logic NOT gate and the corresponding output will be “$\overline{F}$”.
• The inputs “E” and “$\overline{F}$” are connected to logic AND gate and the corresponding output will be ($E\overline{F}$).
• Finally, the outputs $A\overline{B}$,  $C\overline{D}$ and $E\overline{F}$ are connected to a logic OR gate whose output will be $Y=(A\overline{B}+C\overline{D}+E\overline{F})$.

Therefore, the Boolean expression for the given logic circuit is $Y=(A\overline{B}+C\overline{D}+E\overline{F})$.