Solution

Correct answer: P ∨ Q

• In boolean logic, the negation of a variable P is denoted as P̅, and the logical operators "and" (∧) and "or" (∨) have specific meanings. The "and" operator is true only when both operands are true, and the "or" operator is true when at least one of the operands is true.
• The expression (P̅ ∧ Q) ∨ (P ∧ Q̅) ∨ (P ∧ Q) can be thought of as a series of conditions separated by the "or" operator. The first condition (P̅ ∧ Q) states that P is false and Q is true. The second condition (P ∧ Q̅) states that P is true and Q is false. The third condition (P ∧ Q) states that both P and Q are true.
• Since the "or" operator is true when at least one of its operands is true, the entire expression is true whenever any one of these conditions is true. That is, the expression is true when either P is false and Q is true, or when P is true and Q is false, or when both P and Q are true.
• On the other hand, the equivalent expression P ∨ Q states that either P is true or Q is true, or both.
• This is exactly the same as the original expression, since whenever either P is false and Q is true, or when P is true and Q is false, or when both P and Q are true, either P is true or Q is true (or both).
• we can use the truth table method. A truth table lists all possible combinations of truth values (true or false) for the variables in the expression and the resulting value of the expression

Here's the truth table for the original expression:

P  Q  P̅     (P̅ ∧ Q)        Q̅              (P ∧ Q̅)        (P ∧ Q)              (P̅ ∧ Q) ∨ (P ∧ Q̅)                   (P̅ ∧ Q) ∨ (P ∧ Q̅) ∨ (P ∧ Q)
-----------------------------------------------------------------------------------------------------------------------------------------------------------
T  T    F         F               F                  F               T                                     F                                                 T
T  F    F         F               T                  T               T                                     T                                                 T
F  T    T         T               F                  F               F                                     T                                                 T
F  F    T         F               T                  F               F                                     F                                                 F

P  Q  (P̅ ∧ Q) ∨ (P ∧ Q̅) ∨ (P ∧ Q)
--------------------------------------
T  T    T
T  F    T
F  T    T
F  F    F

And here's the truth table for the equivalent expression P ∨ Q:

P  Q  P ∨ Q
--------------
T  T    T
T  F    T
F  T    T
F  F    F

As we can see, the two tables are identical, which means that the two expressions are equivalent.