The equation in the image appears to be a Boolean expression simplification problem with multiple-choice answers. The expression given is:

xyz̅ + xy̅z + x̅y̅z̅

We aim to simplify this expression and find which choice among (a), (b), (c), and (d) is equivalent.

Step 1: Write the expression clearly

The given Boolean expression is: xyz̅ + xy̅z + x̅y̅z̅

Step 2: Simplify the expression

1. Group similar terms for factoring:

xyz̅ + xy̅z + x̅y̅z̅

No direct common factors exist across all terms, so let's proceed to analyze by pairs or individual terms.

2. Analyze the structure:

• xyz̅: This term is active when $x = 1$, $y = 1$, and $z = 0$.
• xy̅z: This term is active when $x = 1$, $y = 0$, and $z = 1$.
• x̅y̅z̅: This term is active when $x = 0$, $y = 0$, and $z = 0$.

3. Check choices for equivalence:

• (a) (x + y)z̅: Expanding this: (x + y)z̅ = xz̅ + yz̅ This does not match the given expression.

• (b) xy + y̅z̅: Expanding this: xy + y̅z̅ This does not match either.

• (c) x + xy̅z̅: Expanding this: x + xy̅z̅ This does not match the original expression.

• (d) z̅(xy + x̅y̅): Expanding this: z̅(xy + x̅y̅) = xyz̅ + x̅y̅z̅ However, this is incomplete. Upon re-checking, no solution matches directly.

Can simplify directly againLet's go back to carefully analyze and simplify the original Boolean expression. The given Boolean expression is:

xyz̅ + xy̅z + x̅y̅z̅

We aim to simplify this to match one of the multiple-choice options. Here is the systematic approach:

Step 1: Analyze each term

• xyz̅: True when $x = 1$, $y = 1$, and $z = 0$.
• xy̅z: True when $x = 1$, $y = 0$, and $z = 1$.
• x̅y̅z̅: True when $x = 0$, $y = 0$, and $z = 0$.

Step 2: Check pairwise terms for simplifications

Let’s compare terms to find a factorization.

Combine xyz̅ and xy̅z:

xyz̅ + xy̅z = x(yz̅ + y̅z) Notice that yz̅ + y̅z = z \oplus y (Exclusive-OR). Thus, we can write: xyz̅ + xy̅z = x(y \oplus z)

So, the first two terms simplify to: $x(y \oplus z)$

Now include x̅y̅z̅:

The full expression becomes: x(y \oplus z) + x̅y̅z̅

Step 3: Match the multiple-choice options

Option (a): (x + y)z̅

Expanding: (x + y)z̅ = xz̅ + yz̅ This does not match the original expression.

Option (b): xy + y̅z̅

Expanding: xy + y̅z̅ This also does not match the original expression.

Option (c): x + xy̅z̅

Expanding: x + xy̅z̅ This does not match the original expression either.

Option (d): z̅(xy + x̅y̅)

Expanding: z̅(xy + x̅y̅) = xyz̅ + x̅y̅z̅ This matches part of the original expression but not fully.

Final Answer: (d) z̅(xy + x̅y̅) is the closest simplified form.

Let me know if you need further clarifications! Here are some questions to consider:

1. What is the importance of Boolean algebra in digital circuits?
2. How does Exclusive-OR ($\oplus$) simplify Boolean expressions?
3. Can Boolean simplifications always be uniquely solved?
4. Why do truth tables help verify Boolean expressions?
5. How can Boolean expressions be implemented in hardware?

Tip: Always verify simplifications with a truth table to ensure correctness.