Boolean Expression Simplification: A + A' + B'

Understanding Boolean algebra is fundamental in digital electronics and logic design. This question asks us to simplify the Boolean expression A + A' + B' using basic Boolean identities.

Let's break down the simplification process step-by-step:

1. Identify the Expression: The given Boolean expression is \( \text{A + A' + B'} \).
2. Apply the Complement Law: One of the core Boolean identities is the Complement Law, which states that any Boolean variable ORed with its complement always results in 1.
   • In mathematical terms, \( \text{A + A'} = \text{1} \).
   • This identity means that either A is true (1) or its complement A' is true (1), so their OR combination will always be true (1).
3. Substitute into the Expression: Now, substitute \( \text{A + A'} \) with 1 in the original expression:
   • \( (\text{A + A'}) + \text{B'} \) becomes \( \text{1 + B'} \).
4. Apply the Dominance Law (OR with 1): Another important Boolean identity is the Dominance Law (also known as the Identity Law for ORing with 1), which states that anything ORed with 1 always results in 1.
   • In mathematical terms, \( \text{1 + X} = \text{1} \), where X can be any Boolean variable or expression.
   • In our case, \( \text{X} \) is \( \text{B'} \). Therefore, \( \text{1 + B'} = \text{1} \).
   • This identity signifies that if one part of an OR operation is already true (1), the entire outcome will be true (1), regardless of the state of the other variable.
5. Final Simplified Expression: Following these steps, the expression \( \text{A + A' + B'} \) simplifies to 1.

In summary, the simplification is as follows:

• \( \text{A + A' + B'} \)
• \( (\text{A + A'}) + \text{B'} \) (using associativity)
• \( \text{1 + B'} \) (since \( \text{A + A'} = \text{1} \))
• \( \text{1} \) (since \( \text{1 + X} = \text{1} \))

Therefore, the equivalent of the Boolean expression \( \text{A + A' + B'} \) is 1.