You wish to prove that three propositions p1, p2, and p3 are equivalent. will it suffice to show that p1 --> p2, p2 --> p3, and p3 --> p1? justify your answer

It is sufficient to prove that  

Step-by-step explanation:

The propositions being equivalent means they should always have the same truth value. If one of them is true, then all of them must be true. And if one of them is false, then all of them must be false.

Suppose we've proven that (call these first, second and third implications).

If was true, then by the first implication that we proved, it would follow that is also true. And then by the second implication that we prove it would follow then that is also true. Therefore the three of them would be true. Notice the reasoning would have been the same if we had started assuming that the one that was true was either . So one of them being true makes all of them be true.

On the other hand, if was false, then by the third implication that we proved, it would follow that has to be false (otherwise would have to be true, which would be a contradiction). And then, since is false, by the second implication that we proved it would follow that is false (otherwise would have to be true, which would be a contradiction). Therefore the three of them would be false. Notice the reasoning would have been the same if we had started assuming that the one that was false was either . So one of them being false makes all of them be false.

So, the three propositions always have the same truth value, and so they're all equivalent.

0.16 or 16%

step-by-step explanation:

lets take a coin for example.

if you toss it twice, it has a 1/4 chance of landing on heads both times because 1/2 * 1/2 is 1/4.

with the cards, the chance of picking dog is 0.4. having two people pick one is like picking a card twice in a row. so 0.4 * 0.4 is 0.16, or 16%