Discrete Math Help

1. The problem statement, all variables and given/known data

Use the logical equivalences [tex] p \rightarrow q \equiv \sim p \vee q [/tex] and [tex] p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p) [/tex] to rewrite the statement form:

[tex](p \rightarrow (q \rightarrow r)) \leftrightarrow ((p \wedge q) \rightarrow r) [/tex]
2. Relevant equations
None


3. The attempt at a solution

My initial assumption is to set this up as:

Let P == [tex](p \rightarrow (q \rightarrow r))[/tex]
Let Q == [tex]((p \wedge q) \rightarrow r) [/tex]

Then plug in from there with the equivalences to get:

[tex] (p \rightarrow q) \wedge (q \rightarrow p) [/tex]

Furthermore:

[tex] (\sim p \vee q) \wedge (q \rightarrow p) [/tex]

Is this the right approach, or am I starting it out wrong?

 

Heres what I ended up with:

[tex] \sim ( \sim p \vee ( \sim q \vee r)) \vee (( \sim p \wedge q) \vee r) \wedge ( \sim( \sim p \wedge q)\vee r) \vee ( \sim p \vee ( \sim q \vee r))[/tex]