Simplifying the Boolean Expression: a + (a̅ b)

This explanation details the process of simplifying the Boolean expression a + (a̅ b), demonstrating its equivalence to another standard form using fundamental Boolean algebra laws.

Understanding the Goal

The objective is to find a simpler, equivalent form for the given Boolean expression a + (a̅ b). This involves applying known rules of Boolean algebra to manipulate the expression.

Key Boolean Algebra Laws Applied

We will utilize the following laws:

• Distributive Law: $X + (Y \cdot Z) = (X + Y) \cdot (X + Z)$
• Complement Law: $X + X̅ = 1$
• Identity Law: $1 \cdot X = X$

Step-by-Step Simplification

Let's simplify the expression a + (a̅ b):

1. Start with the expression:

   $$a + (a̅ \cdot b)$$

2. Apply the Distributive Law, where $X = a$, $Y = a̅$, and $Z = b$:

   $$ (a + a̅) \cdot (a + b) $$

3. Use the Complement Law ($a + a̅ = 1$):

   $$ 1 \cdot (a + b) $$

4. Apply the Identity Law ($1 \cdot X = X$):

   $$ a + b $$

Conclusion

Through the application of the Distributive Law, Complement Law, and Identity Law, the Boolean expression a + (a̅ b) simplifies to a + b.

This equivalence, a + (a̅ b) = a + b, is a useful identity in Boolean algebra, sometimes referred to as a variation of the absorption law.