Boolean Algebra Laws: Identifying the False Statement

This explanation breaks down the properties of Boolean algebra to identify which of the given statements is incorrect. We will analyze each statement using fundamental Boolean algebra laws.

Understanding Boolean Algebra Laws

Boolean algebra is used in digital logic and computer science. It deals with variables that can have only two values, typically 0 (False) and 1 (True), and uses logical operators like AND (.), OR (+), and NOT (complement, denoted by $\bar{A}$). Key laws include:

• Identity Law: $A + 0 = A$ and $A \cdot 1 = A$
• Complement Law: $A + \bar{A} = 1$ and $A \cdot \bar{A} = 0$
• Distributive Law: $A(B+C) = AB + AC$ and $A+(BC) = (A+B)(A+C)$
• Idempotent Law: $A + A = A$ and $A \cdot A = A$
• Absorption Law: $A + AB = A$ and $A(A+B) = A$
• Derived Law: $A + \bar{A}B = A+B$

Option 1 Analysis: $\bar{A} + B = A$

Let's test this statement. We can use a simple test case. Let $A=0$ and $B=1$. The left side becomes:

$$ \bar{A} + B = \bar{0} + 1 $$

Since $\bar{0} = 1$, this simplifies to:

$$ 1 + 1 = 1 $$

The right side of the statement is $A$, which is $0$ in this case.

Comparing the results: $1 \neq 0$. Therefore, the statement $\bar{A} + B = A$ is not true.

We can also use the derived law $A + \bar{A}B = A+B$. The statement $\bar{A} + B = A$ is not equivalent to any standard Boolean identity.

Option 2 Analysis: $A(\bar{A} + B) = AB$

To verify this statement, we apply the Distributive Law to the left side:

$$ A(\bar{A} + B) = A \cdot \bar{A} + A \cdot B $$

Using the Complement Law ($A \cdot \bar{A} = 0$):

$$ 0 + AB $$

Using the Identity Law ($0 + X = X$):

$$ AB $$

The simplified left side equals the right side ($AB$). Thus, the statement $A(\bar{A} + B) = AB$ is true.

Option 3 Analysis: $CA + C\bar{A}B = CA + CB$

Let's simplify the left side of the equation. First, factor out $C$ using the Distributive Law:

$$ CA + C\bar{A}B = C(A + \bar{A}B) $$

Now, apply the derived law $A + \bar{A}B = A+B$ to the expression inside the parentheses:

$$ C(A + \bar{A}B) = C(A+B) $$

Apply the Distributive Law again:

$$ C(A+B) = CA + CB $$

The simplified left side equals the right side ($CA + CB$). Thus, the statement $CA + C\bar{A}B = CA + CB$ is true.

Conclusion on the False Statement

We have analyzed all the provided Boolean expressions:

• Option 1: $\bar{A} + B = A$ was found to be not true.
• Option 2: $A(\bar{A} + B) = AB$ was found to be true.
• Option 3: $CA + C\bar{A}B = CA + CB$ was found to be true.

Since options 2 and 3 are true, and option 1 is not true, the question asks for the statement that is not true.

Therefore, the statement $\bar{A} + B = A$ is the one that is not true.