Find the complement of Y(a,b)=ab’+a’b, and prove that Y+Y'=1. Give a reason for each step.
--> Y(a,b) = ab’+a’b
--> complementing Y gives
Y'(a,b) = (ab’+a’b)'
= (ab')' . (a'b)' (DeMorgan's law)
= (a' + b).(a+b') (DeMorgan's law)
= a'(a+b') + b(a+b') (Distributive law)
= a'a + a'b' + ab + bb' (Distributive law)
= ab + a'b' (Complement law)
--> Y' = ab + a'b'
--> Y + Y' = ab’+a’b + ab + a'b'
= a(b+b') + a'(b+b')
= a.1 + a'.1 (Complement law)
= a+a' (Identity law)
= 1 (Complement law)
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