Question

In Boolean algebra \( [B, \land, \lor, ', 0, 1] \), the value of \( x' \land (x \lor y) \) is

• A. \( x \land y \)
• B. \( x' \land y \)
• C. \( x' \land y' \)
• D. \( x \land x' \)

Hint:

When simplifying Boolean expressions, it's helpful to remember the order of operations (though less strict than in standard algebra) and to strategically apply the fundamental laws.

Answer 1

The Correct Option is B

Step 1: Apply the Distributive Law.
The distributive law in Boolean algebra states that \( A \land (B \lor C) = (A \land B) \lor (A \land C) \). Applying this to the given expression \( x' \land (x \lor y) \), we treat \( x' \) as \( A \), \( x \) as \( B \), and \( y \) as \( C \):
$$x' \land (x \lor y) = (x' \land x) \lor (x' \land y)$$ Step 2: Apply the Complement Law.
The complement law states that \( x' \land x = 0 \). Substituting this into the result from Step 1:
$$(x' \land x) \lor (x' \land y) = 0 \lor (x' \land y)$$ Step 3: Apply the Identity Law.
The identity law states that \( 0 \lor A = A \). Applying this to the result from Step 2, where \( A \) is \( (x' \land y) \):
$$0 \lor (x' \land y) = x' \land y$$ Thus, the simplified value of \( x' \land (x \lor y) \) is \( x' \land y \).