Question

Define an XOR gate and its significance in binary addition.

Answer 1

An XOR (exclusive OR) gate is a digital logic gate that performs the XOR operation on two binary inputs. It has two inputs (A and B) and one output (Y). The output of an XOR gate is HIGH (1) when the number of HIGH inputs is odd, and it is LOW (0) when the number of HIGH inputs is even.

The truth table for an XOR gate is as follows:

A    B    Y
0    0    0
0    1    1
1    0    1
1    1    0

Now, let's discuss the significance of the XOR gate in binary addition. In binary addition, we often need to add two binary digits (bits). The addition of two bits can result in three possible outcomes: 0, 1, or 2.

When adding two bits (A and B), we can use an XOR gate to obtain the sum (S) and a separate AND gate to obtain the carry (C). Here's how it works:

Sum (S): The XOR gate is used to calculate the sum of the two bits. The output of the XOR gate gives the result of adding the two bits without considering any carry. For example, if A = 1 and B = 0, the XOR gate will output 1, indicating the sum of 1 + 0 = 1.

Carry (C): The AND gate is used to calculate the carry generated by adding the two bits. The output of the AND gate gives the carry that is propagated to the next position in the addition. For example, if A = 1 and B = 1, the AND gate will output 1, indicating that there is a carry of 1 from adding 1 + 1.

Using an XOR gate for sum and an AND gate for carry, we can perform binary addition on multi-bit numbers by cascading these gates across different positions. This forms the foundation of binary arithmetic in digital circuits and computers.

For instance, consider the binary addition of two 4-bit numbers:

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Copy code
   1 0 1 0 (A)
 + 0 1 1 1 (B)
 ---------


Here, you'd use four XOR gates to compute each bit of the sum (S) and three AND gates to compute the carry (C) between each pair of corresponding bits. The carry generated at each position is then propagated to the next higher bit position.

In summary, the XOR gate plays a crucial role in binary addition as it helps to calculate the sum of two bits, which is a fundamental operation in binary arithmetic.