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As the title says, how to prove $p ∧ (q ∧ r) ≡ (p ∧ q) ∧ (p ∧ r)$ without using conjunctional laws?

I did attempt this question on my own, but found myself running into road blocks.

Mark
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  • well with all the same operator, parenthesis can be removed and since it is commutative you are just left with $p\land p=p$ simplification. – zwim Sep 28 '19 at 23:34
  • Have you tried to apply the truth table? – user0102 Sep 28 '19 at 23:42
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    What are "conjunction laws". But this seems trivial If $p \land (q\land r)$ then $p$. And also $q\land r$ so $q$. So $p\land q$. And because of $q\land r$ we have $r$ so with $p$ with have $p\land r$. So we have both $p\land q$ and $p\land r$ so $(p\land q)\land(p\land r)$. So $p\land(q\land r)\implies(p\land q)\land(p\land r)$. And if we have $(p\land q)\land(p\land r)$ we have $p\land q$ so $p$ and we have $p\land q$ so $q$ and $p\land r$ so $r$. So $q\land r$ and $p\land(q\land r)$. So $(p\land q)\land(p\land r)\implies p\land(q\land r)$. – fleablood Sep 28 '19 at 23:44
  • @zwim yes that would be simple. I should have mentioned, our professor is not allowing us to use commutative property. – Mark Sep 29 '19 at 00:06
  • Well if you cannot use any basic tools, I see no other solution than the truth table. – zwim Sep 29 '19 at 00:10
  • $p \wedge q \equiv p \wedge q \wedge p$ – LAGRIDA Sep 29 '19 at 00:25
  • "Well if you cannot use any basic tools, I see no other solution than the truth table." The OP doesn't say we can't use basic tools, just not commutative property and "conjunctive laws". I think the OP should tell us what tools we can use. – fleablood Sep 29 '19 at 00:44
  • https://www3.risc.jku.at/education/courses/ws99/formal/slides/logic/index_19.html Conjunctive laws seem to be commutivity and associativity. – fleablood Sep 29 '19 at 00:46
  • @OP so what rules can you use? – Bram28 Sep 29 '19 at 01:24

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Not sure what conjunctional laws are, but you can check they are the same all eight cases:

______p______q______r______p&(q&r)______(p&q)&(p&r)____

______F______F______F______F_____________F__________

______F______F______T______F_____________F__________

______F______T______F______F_____________F__________

______F______T______T______F_____________F__________

______T______F______F______F_____________F__________

______T______F______T______F_____________F__________

______T______T______F______F_____________F__________

______T______T______T______T_____________T__________

Tim
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