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:

$$\sim ( \sim p \vee ( \sim q \vee r)) \vee (( \sim p \wedge q) \vee r) \wedge ( \sim( \sim p \wedge q)\vee r) \vee ( \sim p \vee ( \sim q \vee r))$$