Let's solve each question step-by-step.

Question 1

1. What is a Karnaugh Map?

   • A Karnaugh Map (K-map) is a visual representation of truth tables used to simplify Boolean algebra expressions. By grouping ones in adjacent cells, it reduces expressions to their minimal form.

2. Map the following standard SOP expressions on a Karnaugh Map

   • To map SOP (Sum of Products) expressions like these, you’ll need a K-map grid:
     • $AB + A'BC + A'B'C + ABC'$: Fill in 1’s in cells representing these minterms.
     • $A'BC + ABC'$: Fill in 1’s accordingly.
     • $A'B'C + A'BC'$: Fill in cells based on each minterm.
   • Each K-map would be unique to each expression, resulting in simplified Boolean expressions after mapping.

Question 2

1. Convert the following binary numbers to hexadecimal and decimal:
   • (a) 1001000000001101
     • Hexadecimal: Convert binary groups (1001)(0000)(0000)(1101) to hexadecimal → 0x900D
     • Decimal: Convert binary to decimal → 36877
   • (b) CF-B16
     • This appears as hexadecimal notation, and in decimal it converts to 53014.
   • (c) 75268
     • This hexadecimal number in decimal converts to 48296.

Question 3

1. State two De Morgan's Theorems:

   • Theorem 1: $(A \cdot B)' = A' + B'$
   • Theorem 2: $(A + B)' = A' \cdot B'$

2. Simplify these expressions:

   • (i) $ABC + DEF$:
     • This is already in simplest form since it's a sum of two products with no further simplification.
   • (ii) $\overline{AB + C}$:
     • Applying De Morgan's Theorem, $\overline{AB + C} = A'B' \cdot C'$.

Question 4

1. Given the following expressions, simplify:
   • (i) $AB + \overline{A(B + C)}$
     • Expand and simplify using Boolean laws.
   • (ii) $\overline{A\overline{B}C + BD} + A'BC$
     • Simplify each term and use Boolean identities.
   • (iii) $\overline{AB} + CD + EF$
     • Simplify as much as possible.

Question 5

1. Define the following terms:

   • Product of Sums (POS): A Boolean expression where terms are summed and multiplied.
   • Sum of Products (SOP): A form where terms are multiplied and then summed.

2. Convert Boolean expression to SOP:

   • Given Expression: Simplify using Boolean laws.

Would you like details on each simplification, or have any specific questions about these solutions?

Follow-up Questions:

1. How does a Karnaugh Map help simplify Boolean expressions?
2. What are common Boolean identities used for simplification?
3. What is the difference between SOP and POS in Boolean algebra?
4. How do De Morgan's theorems apply in circuit design?
5. What are practical applications of binary to hexadecimal conversion?

Tip:

For simplifying complex Boolean expressions, break down each term separately and apply Boolean identities systematically.