Binary Logic

Let's simplify the given Boolean expression step-by-step using the laws of Boolean algebra.
Expression:

Step 1: Simplify the innermost parenthesis. Using the distributive law, we can factor out A:
By the law of complementation, . So, .

Step 2: Substitute the simplified part back into the expression. The expression now becomes:

Step 3: Apply the distributive and absorption laws. (since ) By the absorption law, .
So the expression simplifies to:

Step 4: Apply the absorption law again. The expression is . We know . So we have .

Alternatively, . We know . So we have . The final simplified form is .

Let's break down the description of the gate's behavior.

1. "Inhibited by sending a low into one of the inputs": This means if any input is LOW (logic 0), the output is forced to a specific, fixed state, regardless of the other inputs.

2. "the output is HIGH": This fixed state is HIGH (logic 1).

So, we are looking for a logic gate where if any input is 0, the output is always

1. Let's test the options for a 2-input gate with inputs A and B. We'll set B=0 and see what happens to the output. This perfectly matches the description of a NAND gate.

Let's examine the operations:

1. (Exclusive-OR) is defined as:

-

2. (Exclusive-NOR) is defined as:

- Now, let's evaluate each option:

- Option (a): is valid because it is one of the properties of the Exclusive-OR and Exclusive-NOR gates.

- Option (b): is valid, as this is true by definition of XOR and XNOR gates.

- Option (c): is valid. This is a property of XOR.

- Option (d): is incorrect because the left-hand side is 1 (since ) and the right-hand side simplifies to 0 (since ).

Thus, the correct answer is .

We are given the Boolean expression: Let's break this down: 1. XOR operation ( ): This will be true if and have opposite values. 2. Implication ( ): This will be true unless and . Now let's evaluate the truth table for each option:

- Option (a): - (since ) - (since )

- The result is

- Option (b): - (since ) - (since and )

- The result is

- Option (c): - (since ) - (since )

- The result is

- Option (d): - (since ) - (since )

- The result is

Thus, the correct answer is .