Question

In Boolean algebra, if \( a \land x = b \land x \) and \( a \land x' = b \land x' \), then which of the following statements is correct ?

• A. \( a>b \)
• B. \( a<b \)
• C. \( a = b^2 \)
• D. \( a = b \)

Hint:

This problem demonstrates the property of cancellation in Boolean algebra under certain conditions. Multiplying (ANDing) with a variable and its complement can help in proving equalities.

Answer 1

The Correct Option is D

We are given two equations:
1. \( a \land x = b \land x \)
2. \( a \land x' = b \land x' \)
We can use the distributive law and the properties of Boolean algebra to prove \( a = b \).
Consider \( a = a \land 1 = a \land (x \lor x') \) (using \( x \lor x' = 1 \))
Using the distributive law: \( a = (a \land x) \lor (a \land x') \)
From the given equations, we can substitute \( b \land x \) for \( a \land x \) and \( b \land x' \) for \( a \land x' \):
\( a = (b \land x) \lor (b \land x') \)
Using the distributive law in reverse: \( a = b \land (x \lor x') \) Since \( x \lor x' = 1 \), we have:
\( a = b \land 1 \)
Using the identity law \( b \land 1 = b \): \( a = b \)
Therefore, if \( a \land x = b \land x \) and \( a \land x' = b \land x' \), then \( a = b \).