Understanding the XNOR Gate Expression

The question asks to identify the correct logic expression that represents an XNOR gate. Digital logic gates are fundamental building blocks of digital circuits. Each gate performs a specific logical operation on binary inputs (0 or 1) to produce a single binary output.

What is an XNOR Gate?

An XNOR gate, also known as the exclusive-NOR gate, is a digital logic gate that outputs 1 when its inputs are the same (both 0 or both 1) and outputs 0 when its inputs are different. It is the complement or negation of the XOR gate.

Analyzing the Options

Let's examine each given option to determine which one correctly represents the XNOR gate:

• Option 1: \(Y=\overline{A+B}\)

  This expression represents the negation of an OR operation. \(A+B\) is the OR operation, and the bar symbol (\(\overline{}\)) indicates negation. Therefore, \(Y=\overline{A+B}\) is the expression for a NOR gate.

• Option 2: \(Y=\overline{A.B}\)

  This expression represents the negation of an AND operation. \(A.B\) (or AB) is the AND operation, and the bar symbol (\(\overline{}\)) indicates negation. Therefore, \(Y=\overline{A.B}\) is the expression for a NAND gate.

• Option 3: \(Y=\overline{A\odot B}\)

  The symbol \(\odot\) is often used to represent the XNOR operation itself. If \(\odot\) means XNOR, then \(\overline{A\odot B}\) would represent the negation of XNOR, which is the XOR operation. This notation is unusual for the XNOR expression itself.

• Option 4: \(Y=\overline{A\oplus B}\)

  The symbol \(\oplus\) represents the XOR (Exclusive-OR) operation. The expression \(A\oplus B\) gives the output of an XOR gate. The bar symbol (\(\overline{}\)) represents negation. Therefore, \(Y=\overline{A\oplus B}\) represents the negation of the XOR operation. As established earlier, the XNOR gate is the negation of the XOR gate.

Conclusion

Based on the analysis of the options and the definition of the XNOR gate, the expression that correctly represents the XNOR gate is \(Y=\overline{A\oplus B}\).

Revision Table: Basic Logic Gate Symbols and Expressions

Additional Information on XNOR Gate Properties

The XNOR gate has several interesting properties and alternative expressions:

• The output is true (1) if and only if the inputs are equal.
• It is also known as the "equivalence gate" because its output is 1 when the inputs are logically equivalent.
• The expression \(Y = A \odot B\) is a common symbol for XNOR, where \(\odot\) signifies the XNOR operation directly.
• Another way to express the XNOR function using basic gates is \(Y = AB + \overline{A}\overline{B}\). This expression clearly shows that the output is 1 if A and B are both 1 (AB) OR if A and B are both 0 (\(\overline{A}\overline{B}\)).
• The relationship \(XNOR = \overline{XOR}\) is fundamental to understanding the gate.

Understanding these different representations helps in analyzing and designing digital circuits.