Logic Gates

To convert a decimal number into binary, we divide the number by 2 and record the remainders until we get a quotient of 0. The binary equivalent is the sequence of remainders read from bottom to top.
1. Decimal number 21 to binary:
- remainder
- remainder
- remainder
- remainder
- remainder
Reading the remainders from bottom to top, we get the binary equivalent of 21 as: 2. Decimal number 43 to binary:
- remainder
- remainder
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- remainder
- remainder
- remainder
Reading the remainders from bottom to top, we get the binary equivalent of 43 as: Thus, the binary equivalents are: - 21 (decimal) = 10101 (binary) - 43 (decimal) = 101011 (binary)

Step 1: Understanding XOR gate.
The XOR (exclusive OR) gate is a digital logic gate that outputs true (1) if and only if the inputs are different. If both inputs are the same, the output is false (0).
Step 2: Explanation of each option.
- (A) OR gate: Incorrect. The OR gate outputs true if at least one of the inputs is true, regardless of whether they are the same or different.
- (B) AND gate: Incorrect. The AND gate outputs true only if both inputs are true.
- (C) NOT gate: Incorrect. The NOT gate inverts the input but does not depend on whether the inputs are the same or different.
- (D) XOR gate: Correct. The XOR gate outputs true only if the inputs are different.
Step 3: Conclusion.
The correct gate whose output is true only if the inputs are different is the XOR gate.

Step 1: NAND gate logic.
A NAND gate is a NOT AND gate, meaning it outputs the opposite of an AND gate. The boolean expression for an AND gate is: The NOT operation is then applied to this output. Thus, the Boolean expression for the NAND gate is: Step 2: Conclusion.
Hence, the correct Boolean expression for a NAND gate is , which simplifies to .