Hi guys A question regarding propositional logic.
¬(¬P∧Q)∨(P∧¬R)
= (P∨¬Q)∨(P∧¬R) ...DeMorgan's, Double Negation law
= ((P∨¬Q)∨P)∧((P∨¬Q)∨¬R) ...Distribution law
= (P∨¬Q)∧((P∨¬Q)∨¬R) ...Commutative,Associative law
= P∨¬Q ...Absorption law
Keep in mind that the answer is given but I am trying to understand how they got the answer
My issue is with the distribution law. How do you distribute (P∨¬Q)∨(P∧¬R) ?
When multiplying I get the following:
(PvP) V (P ^ !R) V (!Q V R) V (!Q ^ R)
I cant for the life of me figure out what I am doing wrong.
Am I missing a step?
Thank you guys
Denote $\vee:= \cdot$, $\wedge:=+$, $P:=x$, $\neg Q:= y$ and $\neg R := z$, then:
$$\begin{align} (P\vee \neg Q)\vee (P\wedge \neg R) & =(x\cdot y)\cdot(x+ z) \\ & =((x\cdot y)\cdot x)+ ((x\cdot y)\cdot z) \\ & =((P\vee \neg Q)\vee P)\wedge ((P\vee \neg Q)\vee \neg R) \end{align}$$
(This notation should remind you operations in $\mathbb{R}$).