Find the complement of the following expression using dual of a function: 1. xy' + x'y 2. (AB' + C) D' + E 3. AB (C'D + CD') + A'B' (C' +D) (C + D') 4. (x + y+ z) (x' + z') (x + y)

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$1.[xy'+x'y]' = (xy')' . (x'y)' = (x'+y).(x+y') = xy + x'y'$

$2.[(AB'+C)D'+E]' = [(AB'+C)D']'.E' \\= [(AB'+C)'+D] . E'= [(A'+B).C' +D].E' \\ = (A'+B+D).(C'+D).E'$

$3. [AB (C'D + CD') + A'B' (C' +D) (C + D')]'\\=(AB (C'D + CD'))'.(A'B' (C' +D) (C + D'))' \\=(A+B+CD'+DC').(CD+C'D'+A'+B')$

$4.[(x+y'+z)(x'+z')(x+y)]' = (x+y'+z)'+(x'+z')'+(x+y)'\\= x'yz' + xz + x'y$