In the given figure, find the value of (X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) to realise the function F = A + B̅

Explanation:

Given Problem:

The problem requires finding the values of (X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) to realize the function F = A + B̅. The correct option is provided as Option 2.

Step-by-Step Solution:

Understanding the Function:

The given function F = A + B̅ is a logical OR operation between A and the complement of B (denoted as B̅).

• In Boolean algebra, the OR operation (denoted by '+') results in a value of 1 if either of the inputs is 1.
• The complement B̅ represents the inverse of B; if B = 0, then B̅ = 1, and if B = 1, then B̅ = 0.

Truth Table:

To realize this function, we construct a truth table for the inputs A and B and the output F:

Mapping Output to (X₀, X₁, ..., X₇):

The values of (X₀, X₁, ..., X₇) correspond to the output F for all possible combinations of 3-bit binary inputs. For inputs A and B, the truth table above gives the first two bits. The third bit (C) is irrelevant for the function F = A + B̅, meaning the output F remains the same regardless of the value of C.

In a 3-bit binary system:

• X₀ corresponds to A = 0, B = 0, C = 0 → F = 1
• X₁ corresponds to A = 0, B = 0, C = 1 → F = 1
• X₂ corresponds to A = 0, B = 1, C = 0 → F = 0
• X₃ corresponds to A = 0, B = 1, C = 1 → F = 0
• X₄ corresponds to A = 1, B = 0, C = 0 → F = 1
• X₅ corresponds to A = 1, B = 0, C = 1 → F = 1
• X₆ corresponds to A = 1, B = 1, C = 0 → F = 1
• X₇ corresponds to A = 1, B = 1, C = 1 → F = 1

Thus, the values of (X₀, X₁, ..., X₇) are (1, 1, 0, 0, 1, 1, 1, 1), which matches Option 2.

Correct Option:

Option 2: (1, 1, 0, 0, 1, 1, 1, 1)

This option correctly represents the output values for the function F = A + B̅ based on the mapping of the truth table.

Important Information

To further understand why other options are incorrect, let’s analyze them:

Option 1: (1, 1, 0, 0, 1, 1, 1, 1)

This option represents the output values for the function F = A + B̅, but it is identical to Option 2. Therefore, it is redundant and not the correct answer.

Option 3: (1, 1, 0, 1, 0, 1, 1, 1)

This option misrepresents the output values of F. For example:

• X₃ (A = 0, B = 1, C = 1) is listed as 1, but based on F = A + B̅, the correct value should be 0.
• X₄ (A = 1, B = 0, C = 0) is listed as 0, but the correct value is 1.

Thus, Option 3 does not match the truth table of F = A + B̅ and is incorrect.

Option 4: (0, 0, 1, 1, 1, 1, 1, 1)

This option completely misrepresents the output values of F. For example:

• X₀ (A = 0, B = 0, C = 0) is listed as 0, but based on F = A + B̅, the correct value should be 1.
• X₂ (A = 0, B = 1, C = 0) is listed as 1, but the correct value is 0.

Thus, Option 4 does not match the truth table of F = A + B̅ and is incorrect.

Conclusion:

Understanding Boolean functions and their truth tables is critical for solving such problems. The correct values of (X₀, X₁, ..., X₇) for the function F = A + B̅ are (1, 1, 0, 0, 1, 1, 1, 1), which matches Option 2. Other options fail to accurately represent the truth table for the given function, leading to incorrect results.