Prove the equality ac abc ab c a 16 points by truth table

Prove the equality (a.c +a’).(b’+c) = a’.b’ + c

(a) (16 points) by truth table.

(b) (13 points) by Boolean algebra.

(c) (16 points) by proving its dual theorem using Boolean algebra.

Solution

(a) Truth table

(a.c+a\').(b\'+c) = a\'.b\' + c

(b) by Boolean algebra

(a.c +a\') .(b\'+c) = a\'.b\' + c

L.H.S. (a.c +a\') .(b\'+c)

= ab\'c +acc +a\'b\' +a\'c

= ab\'c +ac +a\'b\' + a\'c -------    (cc = c)

= ac(b\' + 1) + a\'b\' + a\'c

= ac + a\'b\' +a\'c ------------(b\'+1 = 1)

= a\'b\' + c(a+a\')

= a\'b\' + c -------(a+a\' = 1)

= a\'c\' +c = RHS

(c) proving dual theorem using Boolean algebra

(a.c + a\') . (b\' + c) = a\'.b\' + c

= ((a\' + c\') . a) + (bc\') = (a + b) .c\'

= (a\'.a + ac\') + bc\' = ac\' +bc\'

= 0 + ac\' +bc\' = ac\' + bc\'

= ac\' +bc\' = ac\' +bc\'

= (a+b)c\' = (a+b)c\'

Hence LHS = RHS