lgbasallote
Evaluate: \[[\neg p \wedge (p \vee q)] \rightarrow q\]



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lgbasallote
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i suppse \[\neg p \wedge (p \vee q) \equiv p\]

lgbasallote
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so then it becomes \[p \rightarrow q\]

lgbasallote
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then what?

lgbasallote
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oh i see

lgbasallote
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i made a mistake

lgbasallote
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\[\neg p \wedge (p \vee q) \equiv F \vee (\neg p \wedge q) \rightarrow q\]

lgbasallote
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then this becomes \[T \wedge (p \vee q) \vee q\]

lgbasallote
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then \[T \wedge (p \vee q) \equiv p \vee q \]

lgbasallote
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in my solution i will be treating them the same....my latex is gonna get confusing if i follow the rules

lgbasallote
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\[\equiv p \vee q \vee q\]
\[q \vee q \equiv T\]
so.. \[\equiv p \vee T \equiv T\]
yes?

lgbasallote
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so then this is a tautology?

PhoenixFire
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implication law is
P>Q=nP V Q
Right?
Sorry for no Latex,

lgbasallote
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yes

lgbasallote
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so is that a yes to my question?

PhoenixFire
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yes it's a tautology, but you messed something up and somehow ended up with a tautology anyways lol

lgbasallote
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where?

UnkleRhaukus
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\[\small¬p∧(p∨q)\Rightarrow q\iff(\neg p\wedge p) \vee(\neg p\lor q)\Rightarrow q\iff (\neg p\lor q)\Rightarrow q \iff (\neg p\Rightarrow q)\vee(q\Rightarrow q )\]\[\iff q\Rightarrow q\qquad \top\]

lgbasallote
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hmm looks different...

lgbasallote
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implication is distributive?

PhoenixFire
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lgbasallote
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seems you're the one who went wrong @PhoenixFire ....

lgbasallote
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\[(\neg p \wedge q) \rightarrow q\]
should become \[\neg(\neg p \wedge q) \vee q\]
by DM law
\[p \vee q \vee q\]

lgbasallote
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hmm seems i missed a step too

lgbasallote
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nevertheless, important thing is.. i was right....that was my question in the first place anyways

PhoenixFire
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de morgans law.... you have to negate both.
n(nP ^ Q) V Q
becomes
nnP V nQ V Q

PhoenixFire
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nnP < Involution law becomes P.
What you missed was distributing the negative to the Q during De Morgan's Law