1. ## Proposition equivalence

Show that the propositions p1 , p2 , p3 , p4 , and p5 can be shown to be equivalent
by proving that the conditional statements p1 -> p4, p3 -> p1, p4 -> p2, p2 -> p5, and
p5 -> p3, are true.

2. Assume that the implications are true. One has to show that $p_i$, $i=1,\dots,5$ are either all true or all false.

Assume that $p_1$ is true; then show that all $p_4, p_2, p_5, p_3$ are true. Now assume that $p_1$ is false; show that $p_3, p_5, p_2, p_4$ are false.

3. Emakarov I appreciate your Help But I need more help in proof. Also Some Other ways.................................

4. So, for example, suppose that the implications are true and p1 is false. We need to show that p3, p5, p2, p4 are false. One of the true implications is p3 -> p1. Looking at the truth table for implication, what does it say about the truth value of p3?

Once you establish that p3 is false, consider the true implication p5 -> p3. What does it say about p5?

To solve this problem, you need to understand when exactly p -> q is true depending on whether p and q are true (the truth table for implication). I also assume that p and q are equivalent, by definition, means that both p and q are true or both are false. If you have a different definition of "equivalent", for example, that p -> q /\ q -> p is derivable in some system, then you need to say so.

5. 1 -> 4
3 -> 1
4 -> 2
2 -> 5
5 -> 3

Show

1 <-> 2
2 <-> 3
3 <-> 4
4 <-> 5
5 <-> 1

(from which it will follow that they're all equivalent, just by going full circle from any one to another).

1 -> 4 and 4 -> 2, so 1 -> 2.
2 -> 5 and 5 -> 3 and 3 -> 1, so 2 -> 1
so 1 <-> 2

Etc.