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Formula is: $((p \wedge q) → r) \wedge (¬(p \wedge q) → r)$

This is what I've already done:

$$((p \wedge q) → r) \wedge (¬(p \wedge q) → r)$$

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

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

And from this point I'm not sure how to proceed. Help would be appreciated.

Sorry, but the last line was written badly(I think). It's fixed now.

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It's really great you got as far as you did; thanks for showing what you've done. –  amWhy Nov 4 '12 at 17:10
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1 Answer 1

up vote 4 down vote accepted

You can continue by using Distributivity of the boolean algebra:

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

$ \Leftrightarrow (¬p \vee ¬q \vee r) \wedge ((p \wedge q) \vee r)$

Here we apply distributivity:

$ \Leftrightarrow (¬p \wedge p \wedge q) \vee (¬q \wedge p \wedge q) \vee (r \wedge p \wedge q) \vee (¬p \wedge r) \vee (¬q \wedge r) \vee (r \wedge r)$

Formally, this is in disjunctive normal form now. We could further simplify:

$ \Leftrightarrow (r \wedge p \wedge q) \vee (¬p \wedge r) \vee (¬q \wedge r) \vee r$

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Sorry for asking, but how did you use distributivity law to get the third line? All I can think of is to use this identity: a∧(b∨c)=(a∧b)∨(a∧c) to get this: ((¬p∨¬q∨r)∧(p∧q))∨((¬p∨¬q∨r)∧r) –  user1242967 Nov 6 '12 at 18:39
    
Here you use: (a∨b∨c)∧(d∨e)=(a∧d)∨(b∧d)∨(c∧d)∨(a∧e)∨(b∧e)∨(c∧e) –  user48415 Nov 6 '12 at 18:59
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The above can be seen as follows: (a∨b∨c)∧(d∨e) = ((a∨b∨c)∧d) ∨ ((a∨b∨c)∧e) = (a∧d)∨(b∧d)∨(c∧d)∨(a∧e)∨(b∧e)∨(c∧e) –  user48415 Nov 6 '12 at 19:06
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