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Assume that only half adders are available in your laboratory. Show that any binary function can be implemented using half adders only.
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But it is partially functionally complete.

$S = A⊕B$

$C = A.B$

We can Perform any operation using Half adder if we can implement basic gates using half adder

AND operation $C = AB$

Not operation $= S$ $($with $A$ and $1$$)$ $= A⊕1 = A'.1+A.1' = A'$

OR operation $= ((A⊕1).(B⊕1))⊕1= (A'.B')' = A+B$
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S + C cannot be done like this rt? As it needs an OR gate
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I mean if any of Led  get ON . we will get S+ C

I think  (A'.B')' will be Ok .

((A⊕1).(B⊕1))⊕1
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Since Not is not a binary function i think it is not required to derive it.

If external 1 is not supplied then also we can get A+B by using two half adders where the inputs of the 2nd half adder will be the Sum S1 and the Carry C1 of the 1st half adder.

So S2=S1⊕ C1=(A ⊕ B) ⊕ (AB) = A+B.

(A ⊕ B) ⊕ (AB)

=(AB'+A'B)(AB)' + (A'B'+AB)(AB)

= (AB'+A'B)(A'+B')+ (0+AB)

=(0+A'B+AB'+0)+AB

= A'B+AB'+AB

= B(A'+A)+AB'

=B+AB'

=A+B

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