I am having trouble reducing a logical expression via logical equivalences. The expression is as follows:
~A~C~D + AB~C~D + ABD + ABC~D + A~B~C~D
Using a Karnaugh Map I am able to reduce the expression to AB + ~C~D.
The problem comes in that I am trying to reduce the expression using logical equivalences. There are two approaches I have tried:
~A~C~D + AB( ~C~D + D + C~D ) + A~B~C~D
~A~C~D + AB( D + ~D(~C + C) ) + A~B~C~D
~A~C~D + AB( ~D + D ) + A~B~C~D
~A~C~D + AB + A~B~C~D
By this method I am able to prove AB, but I cannot see how I would progress to solve ~C~D. The other method is as follows:
~C~D( ~A + AB + A~B) + ABD + ABC~D
~C~D( ~A + A( B + ~B ) ) + ABD + ABC~D
~C~D( ~A + A ) + ABD + ABC~D
~C~D + ABD + ABC~D
By this method I am able to prove ~C~D but I cannot see how I would progress to solve AB.
Is there a way to solve this is one fluid proof that I am not seeing?
Starting from your result
~C~D + ABD + ABC~D
= ~C~D(True + AB) + AB(D + C~D)
= ~C~D + AB~C~D + AB(D+C~D)
= ~C~D + AB(~C~D + C~D + D)
= ~C~D + AB((~C + C)~D + D)
= ~C~D + AB.