# Discrete Math: Proving something is logically equivalent

1. Mar 10, 2012

### sjung915

1. The problem statement, all variables and given/known data
Show that (p ∧ q) → r and (p → r) ∧ (q → r) are not
logically equivalent.

2. Relevant equations
a → b = $\neg$a v b

3. The attempt at a solution
I'm sorry. I'm completely stumped on how to go about this problem. I'm not asking for the solution since I want to know how to do this instead of just getting the answer. Any help would be appreciated. Thank you.
Here is what I had just so no one thinks I didn't try.

(p ∧ q) → r
=> $\neg$ ( p $\wedge$ q ) $\vee$ r
=> ($\neg$p $\wedge$ $\neg$q ) $\vee$ r
=> (switched it around) r $\vee$ ($\neg$p $\wedge$ $\neg$q )
=> (distributed) (r $\vee$ $\neg$p ) $\wedge$ ( r v $\neg$ q)
=> ($\neg$p v r ) $\wedge$ ($\neg$q v r )
=> (p -> r ) $\wedge$ (q -> r)

It said disprove but somehow I'm getting that they are L.E.

Last edited: Mar 10, 2012
2. Mar 10, 2012

### Dick

If you want to prove they are not equivalent then just figure out how you can assign true and false values to p, q and r so that the two sides give you different values.

3. Mar 10, 2012

### sjung915

That makes sense. Here is what I got, please correct me if I'm wrong.

Let
p = true
q = false
r = false
then (p ∧ q) → r is true.
and (p → r) ∧ (q → r) is false.
Hence it's not L.E.

Any mistakes?

4. Mar 10, 2012

### Dick

Looks ok to me. true → false is false. false → false is true.

Last edited: Mar 10, 2012
5. Mar 10, 2012

### Office_Shredder

Staff Emeritus
As far as the original logic goes...

Your problem is here.
not (p and q) = (not p OR not q)

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