Questions: Which of the following best describes the value of the Boolean expression shown below? a ! (b a) (A) The value is always true. (B) The value is always false. (C) The value is true when a has the value false, and is false otherwise. (D) The value is true when b has the value false, and is false otherwise. (E) The value is true when either a or b has the value true, and is false otherwise.

Solution Steps

To determine the value of the Boolean expression \(a \& ! (b || a)\), we need to evaluate it under different combinations of Boolean values for \(a\) and \(b\). We will check all possible combinations of \(a\) and \(b\) (i.e., both true and false) and see which option matches the results.

We need to evaluate the Boolean expression \( a \& ! (b || a) \) for all combinations of \( a \) and \( b \). The possible values for both \( a \) and \( b \) are either \( \text{True} \) or \( \text{False} \).

We compute the expression for each combination:

• For \( a = \text{True}, b = \text{True} \): \[ \text{Result} = \text{True} \& ! (\text{True} || \text{True}) = \text{True} \& \text{False} = \text{False} \]

• For \( a = \text{True}, b = \text{False} \): \[ \text{Result} = \text{True} \& ! (\text{False} || \text{True}) = \text{True} \& \text{False} = \text{False} \]

• For \( a = \text{False}, b = \text{True} \): \[ \text{Result} = \text{False} \& ! (\text{True} || \text{False}) = \text{False} \& \text{False} = \text{False} \]

• For \( a = \text{False}, b = \text{False} \): \[ \text{Result} = \text{False} \& ! (\text{False} || \text{False}) = \text{False} \& \text{True} = \text{False} \]

From the evaluations, we find that the expression \( a \& ! (b || a) \) yields \( \text{False} \) for all combinations of \( a \) and \( b \). Therefore, the value of the expression is always \( \text{False} \).

Final Answer