Math Problem Statement

Write the Boolean expression that represents the combinatorial circuit in Figure 11.36 and write the output of each gate symbolically.

Write the Boolean expression that represents the combinatorial circuit in Figure 11.36 and write the output of each gate symbolically.

Solution

To analyze the given circuit in Figure 11.36 and write the Boolean expression, we will break down the circuit step by step by identifying the logic gates and their operations:

  1. First layer of gates:

    • Two AND gates are present in the first layer:
      • AND1: Inputs xx and yy. Output is A1=xyA_1 = x \cdot y.
      • AND2: Inputs yy and zz. Output is A2=yzA_2 = y \cdot z.
  2. Second layer of gates:

    • OR Gate: Inputs are the outputs from the two AND gates, A1A_1 and A2A_2. Output is O1=A1+A2=(xy)+(yz)O_1 = A_1 + A_2 = (x \cdot y) + (y \cdot z).
  3. Third layer:

    • NOT Gate: Input is zz. Output is N1=¬zN_1 = \neg z.
  4. Fourth layer of gates:

    • AND Gate: Inputs are xx and N1N_1 (the output from the NOT gate). Output is A3=x¬zA_3 = x \cdot \neg z.
  5. Fifth layer (final gate):

    • OR Gate: Inputs are O1O_1 and A3A_3. Output is F=O1+A3=[(xy)+(yz)]+(x¬z)F = O_1 + A_3 = [(x \cdot y) + (y \cdot z)] + (x \cdot \neg z).

Final Boolean Expression:

The final Boolean expression for the circuit is: F=(xy)+(yz)+(x¬z)F = (x \cdot y) + (y \cdot z) + (x \cdot \neg z)

Symbolic Outputs:

  • A1=xyA_1 = x \cdot y
  • A2=yzA_2 = y \cdot z
  • O1=(xy)+(yz)O_1 = (x \cdot y) + (y \cdot z)
  • N1=¬zN_1 = \neg z
  • A3=x¬zA_3 = x \cdot \neg z
  • F=(xy)+(yz)+(x¬z)F = (x \cdot y) + (y \cdot z) + (x \cdot \neg z)

Let me know if you'd like further clarifications or a detailed truth table for this circuit!


Related Questions:

  1. What is the truth table for this circuit?
  2. How does the addition of the NOT gate impact the final output?
  3. Can this Boolean expression be simplified further?
  4. What is the output when x=1,y=1,z=0x = 1, y = 1, z = 0?
  5. How would this circuit behave if x=0,y=0,z=1x = 0, y = 0, z = 1?

Tip:

When analyzing circuits, always work step by step, labeling intermediate outputs clearly to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Boolean Algebra
Logic Gates
Combinatorial Circuits

Formulas

AND operation: A \cdot B
OR operation: A + B
NOT operation: \neg A

Theorems

De Morgan's Laws
Associative, Distributive, and Commutative Laws of Boolean Algebra

Suitable Grade Level

Grades 9-12