Question

Apply DeMorgan's theorem to the following equations: a) \(\mathrm{F}=\mathrm{V}+\mathrm{A}+\mathrm{L}\) b) \(\mathrm{F}=\underline{\mathrm{A}}+\underline{\mathrm{B}}+\underline{\mathrm{C}}+\underline{\mathrm{D}}\) c) \(\mathrm{F}=\mathrm{WXYZ}\) d) \(\mathrm{F}=\underline{\mathrm{ABC}}+\mathrm{D}\)

Short answer

Applying DeMorgan's theorem to the given equations: a) \( F = \overline{V} \cdot \overline{A} \cdot \overline{L} \) b) \( F = A \cdot B \cdot C \cdot D \) c) \( F = \overline{W} + \overline{X} + \overline{Y} + \overline{Z} \) d) \( F = \overline{A} + \overline{B} + \overline{C} + D \)

Step-by-step solution

Understanding DeMorgan's theorem

DeMorgan's theorem states that the negation of a conjunction is the disjunction of the negations. And the negation of a disjunction is the conjunction of the negations. In simply terms : (NOT A AND NOT B) = NOT(A OR B) and (NOT A OR NOT B) = NOT(A AND B).

Applying DeMorgan's theorem to equation a)

To apply DeMorgan's theorem to the equation F = V + A + L, the OR operation (+) between the variables should be changed to the AND operation and each variable should be negated. This results in the equation as follows: \( F = \overline{V} \cdot \overline{A} \cdot \overline{L} \)

Applying DeMorgan's theorem to equation b)

To apply DeMorgan's theorem to the equation F = \(\overline{A}+\overline{B}+\overline{C}+\overline{D}\), each variable needs to be negated and the AND operation used instead of the OR operation. This results in the equation as follows: \( F = A \cdot B \cdot C \cdot D \)

Applying DeMorgan's theorem to equation c)

To apply DeMorgan's theorem to the equation F = WXYZ, the AND operation between the variables should be replaced with the OR operation and each variable should be negated. This will result in the following equation: \( F = \overline{W} + \overline{X} + \overline{Y} + \overline{Z} \)

Applying DeMorgan's theorem to equation d)

To apply DeMorgan's theorem to the equation F = \(\overline{ABC}+D\), replace the AND operation in \(\overline{ABC}\) with the OR operation and negate each variable. This results in the equation as follows: \( F = \overline{A} + \overline{B} + \overline{C} + D \)

Key concepts

Boolean algebra

Boolean algebra is a branch of mathematics that deals with true/false logic operations. It was formulated by George Boole in the 19th century. In Boolean algebra, variables represent truth values—typically 0 and 1, or false and true, respectively. These binary states enable us to perform logical operations like AND, OR, and NOT.
Boolean algebra is foundational in computer science and electrical engineering, especially when working with digital circuits and programming logic. It is used to simplify logical expressions, making complex logic easier to understand and compute.

• AND operation: Only true if both operands are true.
• OR operation: True if at least one operand is true.
• NOT operation: Inverts the truth value.

Understanding Boolean algebra is crucial for applying DeMorgan's Theorem effectively.

Logic gates

Logic gates are the building blocks of digital circuits, using Boolean algebra principles to process binary data. Each gate carries out a basic operation: AND, OR, NOT, NAND, NOR, XOR, and XNOR.
1. AND Gate: Outputs true only if all inputs are true.
2. OR Gate: Outputs true if any input is true.
3. NOT Gate (Inverter): Outputs the opposite value of the input.
4. NAND Gate: Outputs false only if all inputs are true (NOT AND).
5. NOR Gate: Outputs true only if all inputs are false (NOT OR).
Logic gates are diagrammatically represented and combined to execute specific logic functions in hardware, such as generating control signals in microprocessors, or in software, like in algorithms managing decision-making processes.
Learning how these gates operate helps in understanding the transformations seen in DeMorgan's Theorem.

Negation

Negation in Boolean logic refers to the inversion of a truth value. It is denoted with a bar (\( \overline{A} \)) or symbolically as NOT. If a proposition \( A \) is true, then \( \overline{A} \) or NOT \( A \) is false, and vice versa.
This operation is inherently simple yet highly influential in logic expressions and circuit operations. Negation flips a 1 to 0 and vice versa in binary terms.
In DeMorgan's Theorem, negation is a key element as it used to transform the structure of logic expressions:

• The negation of a conjunction (\( A \cdot B \)) becomes the disjunction of negations (\( \overline{A} + \overline{B} \)).
• The negation of a disjunction (\( A + B \)) becomes the conjunction of negations (\( \overline{A} \cdot \overline{B} \)).

Mastering negation is vital for manipulating expressions within Boolean algebra and applying theorems like DeMorgan's.

Conjunction and Disjunction

Conjunction and disjunction are fundamental operations in Boolean algebra. Often represented by AND and OR gates, they define the way statements are connected.
Conjunction (AND operation) is denoted by \( \cdot \) or simply juxtaposition (e.g., \( AB \)). It means both statements must be true for the conjunction to be true. Each component of a conjunction can be considered as a filter that the data must satisfy.
Disjunction (OR operation) uses the symbol +. A disjunction is true if at least one of the connected statements is true. It resembles a logical "either-or" where multiple pathways can lead to a true evaluation.
DeMorgan’s Theorem intricately relates these operations through negation:

• NOT (A AND B) is equivalent to (NOT A) OR (NOT B).
• NOT (A OR B) is equivalent to (NOT A) AND (NOT B).

Understanding these operations is crucial for processing logical statements, optimizing digital circuits, and simplifying complex logical expressions efficiently.