Boolean Expression Simplification Explained

To simplify the given Boolean expression `S = (X + Y) (X + Z)`, we will use fundamental laws of Boolean algebra. Boolean algebra is a branch of algebra in which the values of the variables are the truth values, true and false, usually denoted 1 and 0, respectively.

Understanding the Boolean Expression

The given expression is:

\($S = (X + Y) (X + Z)\)

Here, \(X\), \(Y\), and \(Z\) are Boolean variables, which can take values of either 0 or 1.

Applying Boolean Algebra Laws for Simplification

We can simplify this Boolean expression using a specific Boolean algebra law known as the Distributive Law, or by directly expanding and simplifying using other Boolean identities.

Method 1: Using the Distributive Law (A + BC = (A + B)(A + C))

The distributive law in Boolean algebra has two forms:

• Form 1: `\($A \cdot (B + C) = A \cdot B + A \cdot C\)`
• Form 2: `\($A + (B \cdot C) = (A + B) \cdot (A + C)\)`

Our given expression `\((X + Y)(X + Z)\)` perfectly matches the second form of the distributive law, where `\(A = X\)`, `\(B = Y\)`, and `\(C = Z\)`.

Therefore, by direct application of this law, we can simplify the Boolean expression:

\((X + Y)(X + Z) = X + YZ\)

Method 2: Expanding the Boolean Expression and Simplifying

Alternatively, we can expand the Boolean expression like regular algebraic multiplication and then apply other fundamental Boolean identities step-by-step.

\($S = (X + Y) (X + Z)\)

First, expand the terms:

\($S = X \cdot X + X \cdot Z + Y \cdot X + Y \cdot Z\)

Now, let's apply the following Boolean identities:

• Idempotent Law: `\($A \cdot A = A\)` (So, `\($X \cdot X = X\)` )
• Commutative Law: `\($A \cdot B = B \cdot A\)` (So, `\($Y \cdot X = X \cdot Y\)` )

Substituting these into the expanded expression:

\($S = X + XZ + XY + YZ\)

Next, we can apply the Absorption Law. The absorption law states that `\($A + AB = A\)`.

Consider the terms `\($X + XZ\)` from the expression `\($S = X + XZ + XY + YZ\)`:

\($X + XZ = X(1 + Z)\)

Since `\($1 + Z = 1\)` (Any Boolean variable ORed with 1 results in 1), this simplifies to `\($X \cdot 1 = X\)`.

So, the expression becomes:

\($S = X + XY + YZ\)

Again, apply the absorption law `\($X + XY = X\)` to the first two terms:

\($X + XY = X\)

Therefore, the simplified Boolean expression becomes:

\($S = X + YZ\)

Boolean Simplification Summary

Both methods yield the same simplified form for the Boolean expression `\((X + Y) (X + Z)\)`.

Final Answer Derivation

Based on the simplification using Boolean algebra laws, the expression `\((X + Y) (X + Z)\)` simplifies to `\($X + YZ\)`.