Chapter 4, Problem 64E

Program Plan Intro

Boolean algebra:

• The Boolean expression is known as a mathematical notation that is used to express the function.
• For example: Boolean expression for the NOT gate.

  $\text{X = }\overline{\text{A}}$

Explanation of Solution

Properties of Boolean algebra:

• Six properties of Boolean algebra are shown below:
  • Commutative property
  • Associative property
  • Distributive property
  • Identity property
  • Complement property
  • DeMorgan’s law property
• Commutative property:
  • The commutative property is the property that specifies the production of the same result when adding or multiplying two variables and its reverse order.
  • It is represented in the binary operations with the use of “AND” and “OR” gate.
  • AND operation:
    • Apply the commutative property from left to right or right to left for the given expression using AND operation:

$\text{A}\cdot \text{B = B}\cdot \text{A}$

• For example:
  • Here, take the inputs A as 1 and B as 0 and apply the commutative property for AND operation

$1\cdot 0\text{ = 0}\cdot 1$

$\text{0 = 0}$

• Thus, from the above example, it can be seen that both produce the same result.
• The product of 1 and 0 is 0.
• The product of 0 and 1 is 0.
• OR operation:
  • Apply the commutative property from left to right or right to left for the given expression using the OR operation:

$\text{A+B = B+A}$

• For example: Here, take the inputs A as 1 and B as 0 and apply the commutative property for OR operation:

$1+0\text{ = 0+}1$

$\text{1 = 1}$

• Thus, from the above example, it can be seen that both produce the same result.
• The sum of 1 and 0 is 1.
• Reverse the sum of 0 and 1 is 1.
• Associative property:
  • The associative property is the property that specifies the production of same results when the group of variables is added or multiplied together within the parentheses and its reverse order.
  • It is represented in the binary operations with the use of “AND” and “OR” gate.
  • AND operation:
    • Apply the associative property from left to right or right to left for the given expression using AND operation:

$\text{(A}\cdot \text{B)}\cdot \text{C = A}\cdot \text{(B}\cdot \text{C)}$.

• For example: Here, take the inputs A as 1, B as 0, and C as 1 and apply the associative property for AND operation:

$\text{(1}\cdot 0\text{)}\cdot 1\text{ = 1}\cdot \text{(0}\cdot 1\text{)}$

$\text{0}\cdot 1\text{ = 1}\cdot \text{0}$

$\text{0 = 0}$

• Thus, from the above example, it can be seen that both produce the same result.
• Group of product of A as 1, B as 0, and C as 1 within the parentheses is 0.
• Reverse group of product of A as 1, B as 0, and C as 1 within the parentheses is 0.
• OR operation:
• Apply the associative property from left to right or right to left for the given expression using the OR operation:

$\text{(A+B)+C = A+(B+C)}$.

• For example: Here, take the inputs A as 1, B as 0 and C as 1 and apply the associative property for OR operation:

$\text{(1+0)+1 = 1+(0+1)}$

$\text{1+1 = 1+1}$

$\text{1 = 1}$

• Thus, from the above example, it can be seen that both produce same result.
• Group the sum of A as 1, B as 0, and C as 1 within the parentheses is 1.
• Reverse group of sum of A as 1, B as 0, and C as 1 within the parentheses is 1.

Distributive property:

• The distributive property is represented in the binary operations with the use of “AND” and “OR” gate.
• AND operation:
  • Apply the distributive property from left to right or right to left for the given expression using AND operation:
  • The distributive property is the property when the variable multiplied by a group of variable added together produces the result which is same as that of the variable multiplied separately and then added together.

$\text{A}\cdot (\text{B+C) = (A}\cdot \text{B)+(A}\cdot \text{C)}$.

• For example: Here, take the inputs A as 1, B as 0, and C as 1 and apply the distributive property for AND operation:

$1\cdot (0\text{+1) = (1}\cdot 0\text{)+(1}\cdot 1\text{)}$

$1\cdot 1\text{ = 0+1}$

$\text{1 = 1}$

• Thus, from the above example, it can be seen that both produce same result.
• Sum of 0 and 1 produces the result 1, which when multiplied with 1 produces the result 1.
• Multiply the 1 with 0 separately and multiply 1 with 1 separately and then add both the values which produce the result 1.
• OR operation:
  • Apply the distributive property from left to right or right to left for the given expression using the OR operation:
  • The distributive property is the property when the variable added by a group of variable multiplied together produces the result which is same as that of the variable added separately and then multiplied together.

$\text{A+(B}\cdot \text{C) = (A+B)}\cdot \text{(A+C)}$.

• For example: Here, take the inputs A as 1, B as 0 and C as 1 and apply the distributive property for OR operation:

$\text{1+(0}\cdot 1\text{) = (1+0)}\cdot \text{(1+1)}$

$\text{1+0 = 1}\cdot \text{1}$

$\text{1 = 1}$

• Thus, from the above example, it can be seen that both produce same result.
• Multiply the 0 with 1 produces the result 0, which when added to 1 produces the result 1.
• Sum of 1 and 0 separately and Sum of 1 and 1 separately and then multiply both the values which produce the result 1.
• Identity property:
  • The identity property is the property which produces the same results when sum of 0 and one variable produces the variable itself or product of 1 with one variable produces the variable itself.
  • It is represented in the binary operations with the use of “AND” and “OR” gate.
  • AND operation:
    • Apply the identity property for the given expression using AND operation:

$\text{A}\cdot 1\text{ = A}$.

• For example: Here, take the inputs A as 1 and apply the identity property for AND operation:

$1\cdot 1\text{ = 1}$

$\text{1 = 1}$

• OR operation:
  • Apply the identity property for the given expression using the OR operation:

$\text{A+0 = A}$.

• For example: Here, take the inputs A as 1 and apply the identity property for OR operation:

$\text{1+0 = 1}$.

$\text{1 = 1}$.

• Complement property:
  • The complement property is represented in the binary operations such as “AND” and “OR” gate.
  • AND operation:
    • Apply the complement property for the given expression using AND operation:
    • The product of variable with its complement produces the 0.

$\text{A}\cdot \text{(}\overline{\text{A}}\text{) = 0}$.

• For example: Here, take the inputs A as 1 and apply the complement property for AND operation:

$1\cdot \text{(}\overline{\text{ 1 }}\text{) = 0}$.

$1\cdot 0\text{ = 0}$.

$\text{0 = 0}$

• OR operation:
  • Apply the complement property for the given expression using the OR operation:
  • The Sum of variable with its complement produces the 1.

$\text{A+(}\overline{\text{A}}\text{) = 1}$.

• For example: Here, take the inputs A as 1 and apply the complement property for OR operation:

$\text{1+(}\overline{\text{ 1 }}\text{) = 1}$.

$\text{1+0 = 1}$.

$\text{1 = 1}$.

• DeMorgan’s law property:
  • The DeMorgan’s law property is represented in the binary operations such as “AND” and “OR” gate.
  • AND operation:
    • Apply the complement property for the given expression using AND operation:
    • The DeMorgan’s law states that the complement of results produced in AND gate is equivalent to the complement of the individual inputs and then passed into an OR gate.

$\text{(}\overline{\text{A}\cdot \text{B}}\text{) = }\overline{\text{A}}+\overline{\text{B}}$.

• For example: Here, take the inputs A as 1 and B as 0 and apply the DeMorgan’s law property for AND operation:

$\text{(}\overline{\text{1}\cdot 0}\text{) = }\overline{\text{ 1 }}+\overline{\text{ 0 }}$.

$\text{(}\overline{\text{ 0 }}\text{) = 0}+1$.

$\text{1 = }1$.

• OR operation:
  • Apply the complement property for the given expression using the OR operation:
  • The DeMorgan’s law states that the complement of result produced in OR gate is equivalent to the complement of the individual inputs and then passed into an AND gate.

$\text{(}\overline{\text{A+B}}\text{) = }\overline{\text{A}}\cdot \overline{\text{B}}$.

• For example: Here, take the inputs A as 1 and B as 0 and apply the DeMorgan’s law property for OR operation:

$\text{(}\overline{\text{1+}0}\text{) = }\overline{\text{ 1 }}\cdot \overline{\text{ 0 }}$

$\text{(}\overline{\text{ 1 }}\text{) = 0}\cdot 1$

$\text{0 = 0}$