3.2 Logic Gates and Logic Circuits (3)
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1.
The Boolean expression for a circuit is given as: Y = (A AND B) OR (A AND NOT B).
a) Draw a logic circuit diagram representing this Boolean expression, using the standard logic gate symbols.
b) Construct a truth table to show the output (Y) for all possible input combinations of A and B.
c) Simplify the Boolean expression to a single, equivalent expression. Show your working.
a) Logic Circuit Diagram:
The circuit diagram should consist of:
- An AND gate with inputs A and B.
- A NOT gate with input B.
- An OR gate with inputs the output of the AND gate (A AND B) and the output of the NOT gate (NOT B).
Diagram (cannot be directly rendered in HTML, but would be a visual representation of the described gates connected as described).
b) Truth Table:
| A | B | NOT B | A AND B | A AND NOT B | (A AND B) OR (A AND NOT B) (Y) |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 |
c) Simplified Expression:
Using the distributive law, we can simplify the expression as follows:
Y = (A AND B) OR (A AND NOT B)
Y = A AND (B OR NOT B)
Y = A AND 1
Y = A
2.
A network security system has three conditions: X = "A firewall is enabled", Y = "Antivirus software is installed", and Z = "The operating system is up-to-date". Assume that the system requires all three conditions to be met for full protection. Write a logic expression that represents this requirement. Then, construct a truth table to show the system's protection level for all possible combinations of X, Y, and Z.
The logic expression representing the requirement that all three conditions must be met is X ∧ Y ∧ Z (X AND Y AND Z).
Here's the truth table:
| X | Y | Z | X ∧ Y ∧ Z |
| True | True | True | True |
| True | True | False | False |
| True | False | True | False |
| True | False | False | False |
| False | True | True | False |
| False | True | False | False |
| False | False | True | False |
| False | False | False | False |
The truth table shows that the system provides full protection (True) only when X, Y, and Z are all true.
3.
Consider a combinational logic circuit with two inputs, X and Y, and an output Z. The circuit is designed to implement the Exclusive OR (XOR) function.
a) Provide the Boolean expression for the output Z in terms of X and Y.
b) Draw a logic circuit diagram representing this XOR function using the standard logic gate symbols.
c) Explain, in words, how the XOR gate achieves the XOR function.
a) Boolean Expression:
The Boolean expression for the XOR function is: Z = X XOR Y. This can also be expressed as: Z = (X AND NOT Y) OR (NOT X AND Y).
b) Logic Circuit Diagram:
The circuit diagram should consist of:
- Two AND gates: One with inputs X and NOT Y, and the other with inputs NOT X and Y.
- An OR gate with inputs the outputs of the two AND gates.
Diagram (cannot be directly rendered in HTML, but would be a visual representation of the described gates connected as described).
c) Explanation of XOR Function:
The XOR gate produces a HIGH (1) output only when its two inputs, X and Y, are different. It produces a LOW (0) output when both inputs are the same. This is achieved by the combination of the AND and OR gates. The AND gates ensure that the outputs are HIGH only when one input is HIGH and the other is LOW. The OR gate then combines these outputs, resulting in a HIGH output only when one of the inputs is HIGH and the other is LOW – the XOR function.