Question

Let \( a \) be any element in a Boolean algebra \( B \). If \( a + x = 1 \) and \( ax = 0 \), then:

• A. \( x = 1 \)
• B. \( x = 0 \)
• C. \( x = a \)
• D. \( x = \overline{a} \)

Hint:

In Boolean algebra, the sum \( a + x = 1 \) implies that \( x = \overline{a} \) and the product \( ax = 0 \) ensures that \( a \) and \( x \) are complements.

Answer 1

The Correct Option is D

Step 1: Understanding the problem.
We are given two conditions: 1. \( a + x = 1 \), which means that the sum of \( a \) and \( x \) is 1 in Boolean algebra. 2. \( ax = 0 \), which means that the product of \( a \) and \( x \) is 0 in Boolean algebra.
Step 2: Analyzing the first condition.
The equation \( a + x = 1 \) implies that \( x = \overline{a} \), because in Boolean algebra, if the sum of \( a \) and \( x \) is 1, one of them must be the complement of the other.
Step 3: Analyzing the second condition.
The equation \( ax = 0 \) means that the product of \( a \) and \( x \) is 0. Since \( x = \overline{a} \), we know that \( a \) and \( \overline{a} \) cannot both be 1 at the same time, making their product zero.
Step 4: Conclusion.
Thus, \( x = \overline{a} \), and the correct answer is (d).