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I'm currently answering a logical equivalence question and would like confirmation of the rule I used at one point as it is not in the list of laws, presumably because it's "something you should just know".

Part of my working is (NOT(p) OR NOT(q)) OR r == (NOT(p) OR r) OR (NOT(q) OR r)
to then be simplified later to (p implies r) OR (q implies r)

Constructing a truth table and also constructing it through LogicWorks I believe that the two are logically equivalent, but what is the law I used on the first part? Associative?

EDIT:
I realize that the LHS and RHS may be logically equivalent but could I have missed a step out?

Prasad Jadhav
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user1763046
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1 Answers1

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p => r = (¬p) ∨ r 
q => r = (¬q) ∨ r
(p => r) ∨ (q => r) = ((¬p) ∨ r) ∨ ((¬q) ∨ r)
(p => r) ∨ (q => r) = (¬p) ∨ (¬q) ∨ r ∨ (¬q) ∨ r ∨ (¬p) ∨ r∨r -- distribution
(p => r) ∨ (q => r) = (¬p) ∨ (¬q) ∨ r -- elimination of duplicates
Aubin
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