Question

Argue that the relation "logically implies" is transitive on the set of all statements. That is, if a, b, and c are statements such that a→b, b→c, then c→a

Answer 1

Are you sure it's not a→c?

Answer 2

oops, yeah that is what it's supposed to be!

Answer 3

An example
a = The person has wide hips
b = she is a female
c = she is human

Answer 4

a implies b because generally, only females have wide hips.
b implies c because females are human.
a implies c directly without going through b because a person is a human.

Answer 5

On the other hand, it does not go backward. c does not imply a because not all humans have wide hips.
Got it?

Answer 6

yeah that makes sense. I'm just not sure if i'm allowed to use specific examples or if it has to be general. So i'm not sure how to explain that in a general form

Answer 7

In general, whenever a implies b, the set b has to be greater than or equal to a. Meaning b greater than or equal to c. So since b is contained in c, and a is contained in b, then a is contained in c.
It's just like putting your phone in your backpack and putting your backpack in your car.
You can simply say your backpack is in your car.

Answer 8

\[a \le b \le c \ \then \ a \le c \ \ so \\a \ is \ contained \\in \\c\]

Answer 9

Okay that actually helps alot. thank you!!

Answer 10

I actually meant you can simply say your phone is in your car.
Either way, you're welcome and good luck.

Answer 11

I just realized that this doesnt really show how it is transitive

Answer 12

It does.