For every combination of x,y,z the function value remains same for input x¯,y,z¯

Introduction:
The given problem is related to Boolean expressions and involves the function f(x,y,z) satisfying the given relation f(x¯,y,z¯) = f(x,y,z). We need to determine the maximum number of Boolean expressions that can be formed for this function.

Understanding the Problem:
To understand the problem clearly, let's break it down into smaller parts:
- We have a Boolean function f(x,y,z) that takes three variables x, y, and z.
- The given relation f(x¯,y,z¯) = f(x,y,z) states that the output of the function remains the same when the complement of x and z is taken.
- We need to find the maximum number of Boolean expressions that satisfy this relation.

Approach:
To find the maximum number of Boolean expressions, we can analyze the possible combinations of values for the variables x, y, and z. Let's consider each variable separately:

Variable x:
- We have two possibilities for x: x or x¯ (complement of x).
- Since the complement of x does not affect the output, we can consider x as a constant (0 or 1) for all combinations.

Variable y:
- As y is not involved in the given relation, it can take any value (0 or 1) independently.

Variable z:
- Similar to variable x, we have two possibilities for z: z or z¯.
- Since the complement of z does not affect the output, we can consider z as a constant (0 or 1) for all combinations.

Total Combinations:
- Considering x as a constant, y can take two values (0 or 1), and z as a constant, we have 2 * 2 * 2 = 8 possible combinations.

However, we need to consider that x can also take two values (x or x¯). Hence, the total number of Boolean expressions satisfying the given relation is 8 * 2 = 16.

Conclusion:
Therefore, the correct answer is '16'. The maximum number of Boolean expressions that can be formed for the function f(x,y,z) satisfying the relation f(x¯,y,z¯) = f(x,y,z) is 16.