# Discrete Math: Proving something is logically equivalent

• sjung915
In summary: Your first line should be=> (\negp v \negq) => rYour second line is then=> (r v \negp) ^ (r v \negq)=> (p -> r) ^ (q -> r)Which is the correct answer.In summary, to show that (p ∧ q) → r and (p → r) ∧ (q → r) are not logically equivalent, one can assign true and false values to p, q, and r and observe that they result in different truth values for the two statements. Alternatively, one can use the logical equivalence of ¬(p ∧ q) = (¬p ∨ ¬q) to rewrite the first statement as (¬
sjung915

## Homework Statement

Show that (p ∧ q) → r and (p → r) ∧ (q → r) are not
logically equivalent.

## Homework Equations

a → b = $\neg$a v b

## 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:
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.

Dick said:
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.

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?

sjung915 said:
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?

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

Last edited:
As far as the original logic goes...

sjung915 said:
=> $\neg$ ( p $\wedge$ q ) $\vee$ r
=> ($\neg$p $\wedge$ $\neg$q ) $\vee$ r

not (p and q) = (not p OR not q)

## What is Discrete Math?

Discrete Math is a branch of mathematics that deals with mathematical structures that are countable or finite in nature. It involves studying discrete objects and their properties, such as graphs, sets, and logical statements.

## What does it mean to prove something is logically equivalent?

To prove that two statements are logically equivalent means to show that they have the same truth value in all possible cases. This can be done by constructing truth tables or using logical equivalences and laws.

## What are some common methods used to prove logical equivalence?

Some common methods used to prove logical equivalence include direct proof, proof by contradiction, and proof by induction. These methods involve using logical reasoning and mathematical principles to show that two statements are equivalent.

## Why is proving logical equivalence important in Discrete Math?

Proving logical equivalence is important in Discrete Math because it helps us understand the relationships between different mathematical structures and statements. It also allows us to simplify complex statements and make logical deductions.

## What are some tips for successfully proving something is logically equivalent?

Some tips for successfully proving something is logically equivalent include understanding the basic laws and principles of logic, breaking down complex statements into simpler forms, and practicing with various examples and exercises.

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