SOLUTION: 1. J v (K · L) 2. ~ K //  J

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Question 1205245: 1. J v (K · L)
2. ~ K //  J
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

1. J v (K · L)
2. ~ K    //  J

3. ~ K v ~ L        2, Addition
4. ~ (K · L)        3, DeMorgan's law
5. (K · L) v J      1, Commutation
6. J              5,4, Disjunctive syllogism  

Edwin

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Tutor Edwin offers a great direct proof.
I'll show an indirect proof (aka proof by contradiction).

The idea is to assume the opposite of the conclusion.
From there, show it leads to a contradiction, and hence the original conclusion must be the case.
NumberStatementLine(s) UsedReason
1J v (K & L)
2~K
:.J
3~JAssumption for Indirect Proof
4K & L1, 3Disjunctive Syllogism
5K4Simplification
6K & ~K5, 2Conjunction
7J3 - 6Indirect Proof

Here is the list of the rules of inference and rules of replacement
https://www.algebra.com/algebra/homework/Conjunction/logic-rules-of-inference-and-replacement.lesson