3.2 Logic Gates and Logic Circuits (3)
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1.
Consider the following statements: A = "The weather is sunny", B = "It is raining", and C = "The ground is wet". Write a logic expression that represents the statement "The ground is wet if and only if it is raining or the weather is sunny". Then, determine the truth value of this expression if A is true and B is false.
The logic expression representing "The ground is wet if and only if it is raining or the weather is sunny" is C ↔ (B ∨ A) (C if and only if (B or A)). This expresses the biconditional relationship.
Given that A is true and B is false, we evaluate the expression:
- B ∨ A
- False ∨ True
- True
- C ↔ (B ∨ A)
- C ↔ True
- The biconditional is true only if both sides have the same truth value. Therefore, C must also be true.
Therefore, the statement is true only if C is true. Without knowing the value of C, we cannot definitively say whether the overall expression is true or false. However, for the expression to be true, C must be true.
2.
Question 1
A logic circuit is designed to implement the Boolean expression: Y = (A OR B) AND (NOT C).
Diagram (example)
Draw a logic circuit diagram to represent this expression. Clearly indicate the inputs, outputs, and logic gates used. Include a truth table to verify the circuit's functionality for all possible input combinations.
Answer 1
Circuit Diagram:
The circuit diagram would consist of the following components:
- Two OR gates: One to implement A OR B, and another to implement the OR of the output of the first OR gate and NOT C.
- One NOT gate: To invert the input C.
- One AND gate: To combine the output of the two OR gates.
Truth Table:
| A | B | C | Y |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
This truth table demonstrates that the circuit correctly implements the given Boolean expression for all possible input combinations.
3.
Construct the truth table for a 3-input AND gate. Clearly indicate the inputs (A, B, C) and the output (D). Include all possible input combinations.
Truth Table for a 3-input AND Gate:
| A | B | C | D |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |