Answer to Question #153639 in Discrete Mathematics for Hajer

Question

36. Show that the propositions pi, p2, p3, and p4 can be shown to be equivalent by showing that p1 ↔ p4, P2 ↔ P3, and p1 ↔ p3.

Answer 1

Let $p_1 ↔ p_4, p_2 ↔ p_3,$ and $p_1 ↔ p_3$.

Let $p_1$ is true. Since $p_1 ↔ p_4$ and $p_1 ↔ p_3$, we conclude that $p_4$ and $p_3$ are true. Taking into account that $p_2 ↔ p_3$, we conclude that $p_2$ is also true.

Let $p_1$ is false. Since $p_1 ↔ p_4$ and $p_1 ↔ p_3$, we conclude that $p_4$ and $p_3$ are false. Taking into account that $p_2 ↔ p_3$, we conclude that $p_2$ is also false.

Therefore, the propositions $p_1, p_2, p_3,$ and $p_4$ are equivalent.