Given the following Boolean truth table (A and B are inputs and R is output):

A    B    R

T    T    F

T    F    F

F    T    T

F    F    T

Which of the following Boolean expressions correctly describe the relationship between input and output?

Question

Given the following Boolean truth table (A and B are inputs and R is output):

A    B    R

T    T    F

T    F    F

F    T    T

F    F    T

Which of the following Boolean expressions correctly describe the relationship between input and output?

BlackTom AI · Accepted Answer

We start by restating the problem in our own words: A and B are inputs, R is the output, and the truth table shows:
A B | R
T T | F
T F | F
F T | T
F F | T

Now we scrutinize each option to see which Boolean expression matches this output pattern.
Option 1: R = AB + AB'. This can be simplified using Boolean algebra: AB + AB' = A(B + B') = A(1) = A. So this expression says R = A. Checking the truth table, R would be 1 whenever A is 1 (TT and TF rows), but the table has R = F for both TT and TF. Therefore, this option does not describe the given relationship.
Option 2: R = A'B + A'B'. Factor out A': R = A'(B + B'). Since B + B' = 1, this becomes R = A'. This means R is the complement of A and does not depend on B. Looking at the table, when A is 0 (F), R is 1 for both B values (FT and FF rows), and when A is 1 (T), R is 0 for both B values (TT and TF rows). This exactly matches R = A'. Hence this expression correctly describes the relationship.
Option 3: R = AB + A'B'. This is the XNOR form (A ⇔ B). It yields 1 when A and B are equal, and 0 when they differ. Checking the table: TT would give 1, but the table has R = 0; TF would give 0, but the table has R = 0 (that matches); FT would give 0, but the table has R = 1 (mismatch); FF would give 1, but the table has R = 1 (that matches). Because TT mismatches, this option does not fit the entire truth table.
Option 4: None of these three answers. Since Option 2 provides a correct match to the truth table, this option is not consistent with the observed data.
To summarize the evaluation: Option 1 produces R = A, which contradicts the TT. Option 2 reduces to R = A', which aligns perfectly with the TT. Option 3 yields the XNOR function, which fails on the TT’s TT case. Option 4 is ruled out because a correct option exists among 1–3.

Given the following Boolean truth table (A and B are inputs and R is output): A B R T T F T F F F T T F F T Which of the following Boolean expressions correctly describe the relationship between input and output? 单项选择题

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