Question

If \[\left( {A \oplus B} \right) \wedge \left( {\neg A\Theta B} \right) = A \wedge B\] what should be proper symbol in place of \[ \oplus \] and \[\Theta \] to hold the equation.
A. \[ \wedge \]and \[ \vee \]
B. \[ \wedge \]and \[ \wedge \]
C. \[ \vee \]and \[ \vee \]
D. \[ \vee \]and \[ \wedge \]

Answer 1

Hint: In this question, we need to determine the symbols that represent \[ \oplus \] and \[\Theta \]. For this, first we need to make a truth table for \[A \wedge B\]. For getting the left hand side and right hand side equal, the truth value of both should be true. But this is possible only when one of the inputs is true. At last we will prepare a truth table for the left hand side of the given equation but the change is the symbol \[ \oplus \] is replaced with \[ \wedge \] and the symbol \[\Theta \] is replaced with \[ \vee \].

Complete step by step solution:
We know that \[\left( {A \oplus B} \right) \wedge \left( {\neg A\Theta B} \right) = A \wedge B\]
Now, let us prepare the truth table for \[A \wedge B\]. Consider the following truth table for \[A \wedge B\].

A	B	\[A \wedge B\]
T	T	T
T	F	F
F	T	F
F	F	F

Here, we can say that the output is true only if both the inputs are true.

For the Left hand side to be equal to the right hand side, \[A \oplus B\] should be true and \[\neg A\Theta B\] must be true. Now, we will make a truth table for the left hand side of the given equation but the symbol \[ \oplus \] is replaced with \[ \wedge \] and the symbol \[\Theta \] is replaced with \[ \vee \].

A	B	\[\neg A\]	\[A \wedge B\]	\[\left( {\neg A \vee B} \right)\]	\[\left( {A \wedge B} \right) \wedge \left( {\neg A \vee B} \right)\]
T	T	F	T	T	T
T	F	F	F	F	F
F	T	T	F	T	F
F	F	T	F	T	F

In this way, the left hand side is equal to the right side if the symbol \[ \oplus \] is replaced with \[ \wedge \] and the symbol \[\Theta \] is replaced with \[ \vee \]. This gives, \[\left( {A \wedge B} \right) \wedge \left( {\neg A \vee B} \right) = A \wedge B\]
Therefore, the correct option is (A).
Note: Many students make mistakes in preparing the truth table specifically in determining the truth value of the logic expression. This will not give the desired result.