Question

Need help with a discrete math problem.

Answer 1

Also, the periods at the beginning should be commas: Let P, Q, and R be....

Answer 2

@DebbieG

Answer 3

I figure just try to do a truth table? You know the truth values for p implies q?

Answer 4

I know the single arrow: --> means implies. Does the double arrow mean the same thing?

Answer 5

But yeah, I know how to do a truth table for p implies q

Answer 6

It means if and only if and only if. You can think of it in two ways. Either it means that p implies q AND q implies p, or you can say that it means p and q are equal.

Answer 7

Oops *if and only if

Answer 8

_______ right, but what do the arrows mean that are just pointing one direction but
------ have double lines like that.

Answer 9

________
-------

Answer 10

The arrows literally are "implies"

P implies Q =
\[p \implies q\]

P if and only if Q =
\[p \iff q\]

Answer 11

p q r p-->q p-->r
T T T T T
T F T F T
F T T T T
F F T T T
T T F T F
T F F F F
F T F T T
F F F F T

Answer 12

Lookin good. You might want to add a q --> p line

Answer 13

I need to show that if p-->q and p-->r then q-->p

So

p-->q
T
T
F
T
T
T
F
T

Answer 14

Does that show that because p-->q and p-->r that q-->p though?

Answer 15

Just use the same logic for q ----> p that you did for p --->q same rules apply. For all the rows in which both are true, you can conclude that p if and only if q is also true.

Answer 16

Can probably do these by contradiction too? (I get the truth table approach, but I always find proof by contradiction feels more intuitive, lol).

So for a: assume that P holds but not R.

Since P holds and P->Q, Q holds.
Since Q holds, and Q->R, R holds.
But that is a contradiction. Therefore P cannot be true and not R be true; so P->R.

Answer 17

I think truth table is more introductory is all. I learned it before I even did anything with contradiction.

Answer 18

Ah, gotcha. Well @bbkzr31, if the truth table is making sense to you then stick with it, ignore what I said. :) lol

Answer 19

Yeah, I don't really understand contradiction yet, :/

Answer 20

But how would I use the logic from p-->q to show that q-->p?

Answer 21

Well, p implies q means that we can only judge q if p is true. If p is false, we don't care, we can put p ---> q = true. Same thing for q ---> p. We only can judge p based on the truth of q. If q is false, q --->p is true since we have no way to judge p on an implication we didnt make.

Answer 22

Sort of makes sense

Answer 23

so I put F whenever p is F for q-->p?

Answer 24

Only if q is true and p is false. If p is false then that's fine given q is false. We cant at all judge P without having Q be true.

Answer 25

Can you write the truth values for q-->p?

Answer 26

Wait, we want p-->r

Answer 27

But we also want p if and only if q, meaning at some point we care about q ---> p

Answer 28

Is what I wrote so far correct?

Answer 29

Oh ok

Answer 30

so we first get q-->p and from there we can figure out p-->r

Answer 31

So if we want to only look at p if and only if q

p q p ----> q q----> p
t t t t
t f f t
f t t f
f f f f


So in regards to p if and only if q, it only occurs when both are true or both are false. We can add on more for r, but we don't need to include r for this step.

Answer 32

I meant to put t t at the bottom line

Answer 33

f f t t

my bad

Answer 34

Ah ok, that makes sense

Answer 35

So both are only true in two instances

Answer 36

Correct.

Answer 37

Both must be true or both must be false.

Answer 38

so now we have to show that q-->r?

Answer 39

Right. So we can add on more values like you did before, the 8 combinations of true and false between p q r and use those to show p imp r, q imp r, etc.

Answer 40

and then use those values to find p-->r

Answer 41

Right, usin gthe same kind of logic. We can't judge the truth of r unless is p is true. Therefore if p is false, we can automatically say p ---> r is true. Only if p is true and r is false can we say p ---> r is false. Same thing with q ---> r

Answer 42

Ok, I'm gonna try to write it out real quick

Answer 43

alrighty.

Answer 44

And...stuck on how to finish it :/

Answer 45

Sorry, I gotta head to class x_x Id help ya finish it if I could. Hope I guided ya somewhat, though.

Answer 46

ah ok, well thanks for the help!

Answer 47

yeah ,np :3

Answer 48

I will try to figure the rest out