A new operator, ,or exclusive-or, may be defined by the following truth table Create a propositional calculus expression using only ꓥ, ꓦ and ¬that is equivalent to P Q. Prove this equivalence using truth tables.

The truth table for the exclusive-or operator is:

| P | Q | P ꓥ Q |
|---|---|-------|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |

To create a propositional calculus expression using only ꓥ, ꓦ, and ¬ that is equivalent to P ꓥ Q, we can use the following expression:

(P ꓦ Q) ∧ ¬(P ꓥ Q)

To prove the equivalence using truth tables, we can create a truth table for both expressions and show that they have the same values for all possible truth value combinations of P and Q.

| P | Q | P ꓥ Q | P ꓦ Q | ¬(P ꓥ Q) | (P ꓦ Q) ∧ ¬(P ꓥ Q) |
|---|---|-------|-------|----------|---------------------|
| F | F | F | F | T | F |
| F | T | T | T | F | F |
| T | F | T | T | F | F |
| T | T | F | T | T | F |

As we can see, both expressions have the same values for all possible truth value combinations of P and Q, indicating that they are equivalent.