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Consider the following Boolean function with four variables f(w,x,y,z) = Σ(1,3,4,6,9,11,12,14).
The function is :
- a)Independent of one variables
- b)Independent of two variables
- c)Independent of three variables
- d)Dependent on all variables
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Consider the following Boolean function with four variables f(w,x,y,z...
Given
4-variable function is,
f(w,x,y,z) = Σm(1,3,4,6,9,11,12,14).
Solving above function using K-map in SOP form as,
The minimized function of above K-map is,
So,F depends upon only 2-variables x and z and independent of other 2-variables w and y .
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Community Answer
Consider the following Boolean function with four variables f(w,x,y,z...
Boolean Function Analysis:
To determine the independence of variables in a Boolean function, we need to analyze the function's truth table. Let's first construct the truth table for the given function f(w, x, y, z).
Truth Table:
The function f(w, x, y, z) = Σ(1,3,4,6,9,11,12,14) can be represented by the following truth table:
| w | x | y | z | f(w, x, y, z) |
|---|---|---|---|--------------|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 |
Independence of Variables:
To determine the independence of variables, we need to check if changing the value of any variable affects the output of the function. If the function's output remains the same regardless of the variable's value, it is independent of that variable.
Variable w:
When w is 0, the function takes the values 0, 1, 1, 0, 1, 0, 0, 1 for different combinations of x, y, and z. When w is 1, the function takes the values 1, 0, 1, 1 for different combinations of x, y, and z. Therefore, changing the value of w affects the output of the function. Hence, the function is dependent on the variable w.
Variable x:
When x is 0, the function takes the values 0, 1, 1, 0, 1, 0, 0, 1 for different combinations of w, y, and z. When x is 1, the function takes the values 1, 0, 1, 1 for different combinations of w, y, and z. Therefore, changing the value of x affects the output of the function. Hence, the function is dependent on the variable x.
Variable y:
When y is
To determine the independence of variables in a Boolean function, we need to analyze the function's truth table. Let's first construct the truth table for the given function f(w, x, y, z).
Truth Table:
The function f(w, x, y, z) = Σ(1,3,4,6,9,11,12,14) can be represented by the following truth table:
| w | x | y | z | f(w, x, y, z) |
|---|---|---|---|--------------|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 |
Independence of Variables:
To determine the independence of variables, we need to check if changing the value of any variable affects the output of the function. If the function's output remains the same regardless of the variable's value, it is independent of that variable.
Variable w:
When w is 0, the function takes the values 0, 1, 1, 0, 1, 0, 0, 1 for different combinations of x, y, and z. When w is 1, the function takes the values 1, 0, 1, 1 for different combinations of x, y, and z. Therefore, changing the value of w affects the output of the function. Hence, the function is dependent on the variable w.
Variable x:
When x is 0, the function takes the values 0, 1, 1, 0, 1, 0, 0, 1 for different combinations of w, y, and z. When x is 1, the function takes the values 1, 0, 1, 1 for different combinations of w, y, and z. Therefore, changing the value of x affects the output of the function. Hence, the function is dependent on the variable x.
Variable y:
When y is
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Consider the following Boolean function with four variables f(w,x,y,z) = Σ(1,3,4,6,9,11,12,14).The function is :a)Independent of one variablesb)Independent of two variablesc)Independent of three variablesd)Dependent on all variablesCorrect answer is option 'B'. Can you explain this answer?
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