Which gate is represented by the following truth table? Input Output A B C 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0
NAND
Analyzing the Logic Gate Truth Table
We are given a truth table with three inputs (A, B, C) and one output. Our goal is to determine which logic gate this truth table represents by examining the relationship between the inputs and the output.
| A | B | C | Output |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
Examining the Output of the Truth Table
Looking closely at the output column of the given truth table, we can see a distinct pattern. The output is '1' for every combination of inputs EXCEPT when all three inputs (A, B, and C) are simultaneously '1'. In that specific case (A=1, B=1, C=1), the output is '0'.
Comparing with Standard Logic Gates
Let's compare this behavior to the truth tables of common three-input logic gates:
- AND Gate: A 3-input AND gate's output is '1' only when all three inputs are '1'. Otherwise, the output is '0'. This is the inverse of the given table's pattern. The Boolean expression is \( \text{Output} = A \cdot B \cdot C \).
- OR Gate: A 3-input OR gate's output is '0' only when all three inputs are '0'. Otherwise, the output is '1'. This does not match the given truth table. The Boolean expression is \( \text{Output} = A + B + C \).
- NAND Gate: A 3-input NAND gate is the inverse of an AND gate. Its output is '0' only when all three inputs are '1', and '1' in all other cases. This behavior precisely matches the pattern observed in the given truth table. The Boolean expression is \( \text{Output} = \overline{A \cdot B \cdot C} \).
- NOR Gate: A 3-input NOR gate is the inverse of an OR gate. Its output is '1' only when all three inputs are '0', and '0' in all other cases. This does not match the given truth table. The Boolean expression is \( \text{Output} = \overline{A + B + C} \).
- XOR Gate: A 3-input XOR gate outputs '1' when an odd number of inputs are '1'. This pattern is different from the given table.
- XNOR Gate: A 3-input XNOR gate outputs '1' when an even number of inputs are '1' (including zero). This pattern is also different from the given table.
Identifying the Represented Logic Gate
Based on the comparison, the only standard logic gate whose truth table matches the given table is the 3-input NAND gate. The characteristic feature of the given table is that the output is high (1) unless all inputs are high (1), in which case the output is low (0). This is the defining behavior of a NAND gate.
Revision Table: Comparing Truth Tables
Let's put the given truth table side-by-side with the expected truth table for a 3-input NAND gate for clarity:
| A | B | C | Given Output | 3-Input NAND Output \((\overline{A \cdot B \cdot C})\) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 |
The outputs are identical for all input combinations, confirming that the truth table represents a NAND gate.
Additional Information on Logic Gate Truth Tables
Understanding logic gate truth tables is fundamental to digital electronics. Each gate performs a specific logical function based on its inputs, and the truth table is the complete listing of input-output relationships. NAND gates are considered "universal gates" because any other logic gate (AND, OR, NOT, XOR, NOR, XNOR) can be constructed solely using NAND gates.
Key types of gates include:
- Basic gates: AND, OR, NOT
- Universal gates: NAND, NOR
- Special purpose gates: XOR, XNOR
Analyzing truth tables is a crucial skill for designing and troubleshooting digital circuits.
Important Questions from Logic Gates
Which logical gate is used to represent product of sum expression?
Which logical gate is included in a binary adder circuit to enable binary addition and subtraction?
Which of the following represents the XNOR gate?
A Circuit that operates in such a way that its output is high only when all its inputs are high
______ is also known as Inverter.
