A truth table is used to test whether $\sim(P\leftrightarrow Q)\equiv(P\wedge\sim Q)$. The verdict is:

1. Build both columns over the four combinations of $P$ and $Q$. 2. $P\leftrightarrow Q$ is true when $P$ and $Q$ match, so $\sim(P\leftrightarrow Q)$ is true whenever they differ. 3. That makes $\sim(P\leftrightarrow Q)$ true in two rows: $P$ true with $Q$ false, and $P$ false with $Q$ true. 4. But $P\wedge\sim Q$ is true only in the first of those rows. 5. The columns disagree in the row where $P$ is false and $Q$ is true, so the two are not equivalent. _Source: NECTA ACSEE 2023 Advanced Mathematics 142/2, Question 2: Logic_