Boolean Algebra Simplification: corrected exercises

📚 Introduction: Understanding Sum of Products (SOP)

In the beginning, we must know that this expression is called the Sum of Products (SOP), which is the primary form of any Boolean algebraic expression. We extract it from a truth table, which we will discuss later, and it is also called the canonical form.

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Step 1: Removing Duplicate Terms

In the beginning, we have the expression ABC repeated, so we remove one copy according to the Boolean algebra rule: X + X = X

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Step 2: Identifying Common Factors

In the first two expressions, we have the common factor AB.

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Step 3: Another Common Factor

Then in the last two expressions, we have A as a common factor.

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Step 4: Simplifying Parentheses

After factoring the Boolean algebraic expression, let's focus on the expressions inside the parentheses. We have C + C which equals 1 according to complement law, and the second expression is essentially XNOR between B and C.

🎯 Final Step: Substitution and Result

After substitution, we notice that the expression has become in its abbreviated form. Therefore, the final Boolean algebraic expression is:

📖 Summary of Boolean Algebra Rules Used:

Idempotent Law
X + X = X

Distributive Law
X(Y + Z) = XY + XZ

Complement Law
X + X = 1

XNOR Definition
X ⊕ Y = XY + XY