# How to multiply out a statement form?

I got this form:

(not M or V) and (A or not M) and (not B or M) and (B or V) and (A or not V) and (not A or B)

Or: $$(\neg M\vee V) \wedge (A\vee\neg M) \wedge (\neg B \vee M) \wedge (B\vee V) \wedge (A\vee\neg V)\wedge(\neg A \vee B)$$

Can anyone explain me how to solve it step by step? The result should be this:

A and B and M and V: $$A\wedge B \wedge M \wedge V$$

I dont know how to simplify the statement

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Have you tried "slow and steady" approach? Open one after another... after another... –  Asaf Karagila Dec 18 '12 at 20:38

$$(\neg M\vee V) \wedge \color{blue}{(A\vee\neg M)} \wedge (\neg B \vee M) \wedge \color{red}{(B\vee V) \wedge} \color{blue}{(A\vee\neg V)}\wedge \color{red}{(\neg A \vee B)}\tag{1}$$

$$\color{blue}{[A \lor (\lnot V \land \lnot M)]} \land \color{red}{[B \lor (\lnot A \land V)]} \land \color{green}{(\lnot B \lor M) \land (\lnot M \lor V)}\tag{2}$$

$${[A \lor (\lnot V \land \lnot M)]} \land {[B \lor (\lnot A \land V)]} \land \color{green}{(B \rightarrow M) \land ( M \rightarrow V)}\tag{3}$$

$$\vdots$$

$$A \land B\land M\land V\tag{result}$$

Note: To "deduce" this, I've highlighted some initial steps:

• $(1) \to (2)$ using
• commutativity: $(P \lor Q) \equiv (Q \lor P)$ and $P \land Q \equiv Q \land P$,
• associativity: $P\land (Q \land R) \equiv (P\land Q)\land R$,
• distributive law (applied twice): $(P\lor Q) \land (P\lor R) \equiv P\lor (Q \land R)$, and
• $(2)\to (3)$ using the identity ($\lnot P \lor Q) \equiv (P\rightarrow Q$),
• It follows from $(3)$ (with more work needed to establish) that we must have $A$ and $B$; and since $B$, then also $M$; and since $M$, then also $V$. Why?

It also helps to use a truth-table, from which we can derive the conjunctive-normal form $(\text{result})$ of your expression given in $(1)$:

Note from the truth-table that the given expression evaluates to $\;T=$ true$\;$ only in the first row, if and only if $\;A,\text{ AND}\; B, \text{ AND}\;M, \text{ AND}\; V\;$ are all $\;T=$ true$\;$.
That is, we can conclude:
$$(\neg M\vee V) \wedge {(A\vee\neg M)} \wedge (\neg B \vee M) \wedge {(B\vee V) \wedge} {(A\vee\neg V)}\wedge {(\neg A \vee B)}$$ $$\iff A \land B \land M \land V$$

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But this is not "multiplying out", this is circumventing the algebraic manipulation. If the OP is fine with it, then it's fine. But it might not be the case. (Also congratulation on a fresh new avatar!) –  Asaf Karagila Dec 18 '12 at 21:07
One might call me "not a colorful person", but I was never a big fan of colors... :-P –  Asaf Karagila Dec 19 '12 at 2:28
@Asaf: it simply helped to simplify any necessary explanation/justification. I'm not big on colors, either! –  amWhy Dec 19 '12 at 2:32
I have problems discerning between some of them and they mostly just hurt my eyes. See this meta thread. –  Asaf Karagila Dec 19 '12 at 2:40
+1 for formatting :) –  Parth Kohli Dec 19 '12 at 14:43