Logical Expression Simplification

The question asks us to find the value of the logical expression \(A' \cdot (A' + B')\), where A and B are logical variables. We are given that \(A'\) represents the complement of A, \(A \cdot B\) represents the AND operation, and \(A + B\) represents the OR operation. To solve this, we will use fundamental Boolean algebra laws.

Boolean Algebra Fundamentals

Before we simplify the given expression, let's quickly recall some essential Boolean algebra laws that might be useful:

• Idempotence Law:
  • \(X \cdot X = X\) (for AND operation)
  • \(X + X = X\) (for OR operation)
• Distributive Law:
  • \(X \cdot (Y + Z) = (X \cdot Y) + (X \cdot Z)\)
  • \(X + (Y \cdot Z) = (X + Y) \cdot (X + Z)\)
• Absorption Law:
  • \(X + (X \cdot Y) = X\)
  • \(X \cdot (X + Y) = X\)
• Complement Law:
  • \(X \cdot X' = 0\)
  • \(X + X' = 1\)

Simplifying the Logical Expression \(A' \cdot (A' + B')\)

Let's take the given logical expression: \[A' \cdot (A' + B')\]

We can simplify this expression step-by-step using the Boolean algebra laws.

1. Apply the Distributive Law:

   The Distributive Law states that \(X \cdot (Y + Z) = (X \cdot Y) + (X \cdot Z)\). In our expression, if we consider \(X = A'\), \(Y = A'\), and \(Z = B'\), we can distribute \(A'\) over the terms inside the parenthesis.

   So, \(A' \cdot (A' + B')\) becomes: \[ (A' \cdot A') + (A' \cdot B') \]

2. Apply the Idempotence Law:

   The Idempotence Law for the AND operation states that \(X \cdot X = X\). In our expression, we have \(A' \cdot A'\). Applying this law, \(A' \cdot A'\) simplifies to \(A'\).

   Now, the expression becomes: \[ A' + (A' \cdot B') \]

3. Apply the Absorption Law:

   The Absorption Law states that \(X + (X \cdot Y) = X\). In our current expression, if we consider \(X = A'\) and \(Y = B'\), the form matches \(X + (X \cdot Y)\).

   Therefore, \(A' + (A' \cdot B')\) simplifies directly to \(A'\).

   \[ A' + (A' \cdot B') = A' \]

Final Result and Option Comparison

After simplifying the logical expression \(A' \cdot (A' + B')\) using the laws of Boolean algebra, we found that its value is \(A'\).

Let's compare this result with the given options:

The simplified value \(A'\) matches option 3.