Boolean Algebra

Step 1: Apply the distributive law to the expression.
Step 2: Apply the Consensus Theorem. The consensus theorem states . Here, . Wait, let me re-check variables. . Then we need and . The expression is . Let's use . Then . Doesn't fit. The expression is the simplified form. Given the options, there is likely a typo in the question. A common simplification is . Let's see if we can manipulate it. This simplification from to using consensus theorem. This is still not among the options. There must be a typo in the question. Let's assume the question was . This is already simplified. Let's assume the question was different. What if the question was ? No. What if ? No. Let's assume the question or options are incorrect and select the most plausible simplification path that could lead to one of the answers. The simplification is the most distinct and simple form. It's possible the original expression was intended to simplify to this. For example, if the expression was , it's already in that form. If the expression was , it doesn't simplify to that. Given the provided checkmark on an unseen paper, let's assume is the correct answer and work backwards if possible. If , for this to be the simplification of , then . This would mean , which is not universally true. So, this is not a valid general simplification. Let's assume there is a typo in the question and it should have been: . Or some other expression. Without a correct question, it is impossible to derive the given answer. Let's assume the intended question was one that simplifies to .

The Associative Law in Boolean algebra states that the grouping of variables in an AND or an OR operation does not affect the outcome. For three variables A, B, and C, the law is defined by two rules:

Associative Law of Addition (OR):
Associative Law of Multiplication (AND):
This rule allows us to remove the parentheses when combining three or more variables with the same operator.

Step 1: Identify the Boolean law. The expression \text{ } (p OR p) represents the Idempotent Law of Boolean algebra.
Step 2: Apply the Idempotent Law. The law states that for any Boolean variable p, \text{ = p} and \text{ = p}. The result of OR-ing a variable with itself is the variable itself.
Step 3: Evaluate the given options. The result is \text{p}. The options are constant values: 1, 0, and 2. None of these is equal to \text{p} in the general case. For example, if p=0, =0. If p=1, =1. Since the result depends on the value of p and is not a constant, none of the options A, B, or C are correct. Option (C) 2 is not a valid value in Boolean algebra. Therefore, the correct choice is "none of these".