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With 4 Boolean variables, how many Boolean expressions can be formed?
- a)16
- b)256
- c)1024 (1K)
- d)64 K (64 ×1024)
Correct answer is option 'D'. Can you explain this answer?
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With 4 Boolean variables, how many Boolean expressions can be formed?a...
Understanding Boolean Variables
Boolean variables can take two values: true (1) or false (0). For 4 Boolean variables, each variable can independently be either true or false.
Calculating Possible Combinations
- Each of the 4 variables can have 2 states (0 or 1).
- The total number of combinations for the variables can be calculated using the formula:
\[
\text{Total combinations} = 2^{n}
\]
where \( n \) is the number of variables.
Applying the Formula
- For 4 variables:
\[
\text{Total combinations} = 2^{4} = 16
\]
This represents the number of unique combinations of input values (0s and 1s) for the variables.
Forming Boolean Expressions
- Each combination of variable states can produce a unique Boolean expression.
- A Boolean expression can be formed using logical operations such as AND, OR, and NOT.
Counting the Boolean Expressions
- A given Boolean function can be defined for each of the 16 combinations of input states.
- For each combination, a Boolean expression can either be true or false, leading to \( 2^{16} \) different Boolean functions.
\[
\text{Total Boolean expressions} = 2^{16} = 65,536
\]
This can be simplified to \( 64 \, K \) (or 64 1024).
Conclusion
Thus, the total number of different Boolean expressions that can be formed with 4 Boolean variables is:
- **Correct Answer: 64K (Option D)**.
Boolean variables can take two values: true (1) or false (0). For 4 Boolean variables, each variable can independently be either true or false.
Calculating Possible Combinations
- Each of the 4 variables can have 2 states (0 or 1).
- The total number of combinations for the variables can be calculated using the formula:
\[
\text{Total combinations} = 2^{n}
\]
where \( n \) is the number of variables.
Applying the Formula
- For 4 variables:
\[
\text{Total combinations} = 2^{4} = 16
\]
This represents the number of unique combinations of input values (0s and 1s) for the variables.
Forming Boolean Expressions
- Each combination of variable states can produce a unique Boolean expression.
- A Boolean expression can be formed using logical operations such as AND, OR, and NOT.
Counting the Boolean Expressions
- A given Boolean function can be defined for each of the 16 combinations of input states.
- For each combination, a Boolean expression can either be true or false, leading to \( 2^{16} \) different Boolean functions.
\[
\text{Total Boolean expressions} = 2^{16} = 65,536
\]
This can be simplified to \( 64 \, K \) (or 64 1024).
Conclusion
Thus, the total number of different Boolean expressions that can be formed with 4 Boolean variables is:
- **Correct Answer: 64K (Option D)**.
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With 4 Boolean variables, how many Boolean expressions can be formed?a)16b)256c)1024 (1K)d)64 K (64 1024)Correct answer is option 'D'. Can you explain this answer?
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