Write a proof sequence for the following assertion. Justify one of the steps in

Mathematics
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Apr 7th, 2015

Probably the brace } means logical AND (^). Then, step by step, transform the left part of the implication:

[¬(a ^ ¬b)] ^ [¬b] =          { ¬(X^Y) = (¬X)V(¬Y)  }
[¬a V b] ^ [¬b] =              { (X V Y) ^ Z = (X^Z) V (Y^Z) }
[¬a ^ ¬b] V [b ^ ¬b] =      {  X ^ ¬X,  X AND not X is always false  }
[¬a ^ ¬b] V [false] =        {  (X V false)  equals  X}
¬a ^ ¬b.

Is this implies ¬a?  ¬a ^ ¬b ==> ¬a?

The only situation when  X ==> Y  is false is when  X is true but Y is false.
Here, if the right side ¬a is false, then the left side  (¬a ^ ¬b) = (false ^ ¬b) = false.

So, yes, ¬a ^ ¬b ==> ¬a and therefore the proof is complete.

---------------
(more formally, we can use  (X ==> Y) = (¬X V Y) for the last step, so
(¬a ^ ¬b ==> ¬a) = (¬(¬a ^ ¬b) V ¬a) = (a V b V ¬a) = (b V a V ¬a) = (b V true) = true. )

Apr 8th, 2015

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