Discrete Math Help

  • Thread starter mkienbau
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  • #1
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Homework Statement



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]

Homework Equations


None


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?
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
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You said "Let P= [tex](p \rightarrow (q \rightarrow r))[/tex]" and "Let Q= [tex]((p \wedge q) \rightarrow r) [/tex]" but then wrote "[tex] (p \rightarrow q) \wedge (q \rightarrow p) [/tex]"

Don't you mean [tex] (P \rightarrow Q) \wedge (Q \rightarrow P) [/tex]?

Use [tex] p \rightarrow q \equiv \sim p \vee q [/tex] to rewrite each part of that, then use it again to rewrite P and Q separately and plop them in there.
 
  • #3
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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]
 
Last edited:

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