Discrete Math Help: Rewrite Statement with Logical Equivalences

[mkienbau]

Homework Statement

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

(p \rightarrow (q \rightarrow r)) \leftrightarrow ((p \wedge q) \rightarrow r)

Homework Equations

None

The Attempt at a Solution

My initial assumption is to set this up as:

Let P == (p \rightarrow (q \rightarrow r))
Let Q == ((p \wedge q) \rightarrow r)

Then plug in from there with the equivalences to get:

(p \rightarrow q) \wedge (q \rightarrow p)

Furthermore:

(\sim p \vee q) \wedge (q \rightarrow p)

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

 

[HallsofIvy]

You said "Let P= (p \rightarrow (q \rightarrow r))" and "Let Q= ((p \wedge q) \rightarrow r)" but then wrote "(p \rightarrow q) \wedge (q \rightarrow p)"

Don't you mean (P \rightarrow Q) \wedge (Q \rightarrow P)?

Use p \rightarrow q \equiv \sim p \vee q to rewrite each part of that, then use it again to rewrite P and Q separately and plop them in there.

 

[mkienbau]

Heres what I ended up with:

\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))