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In the expression $\overline{\text{A}}(\overline{\text{A}}+\overline{\text{B}})$ by writing the first term $\text{A}$ as $\text{A + 0}$, the expression is best simplified as

  1. $\text{A+AB}$
  2. $\text{AB}$
  3. $\text{A}$
  4. $\text{A+B}$
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it is given A'(A'+B') in question  which will be simplified to A' irrespective of the method now if we consider that first A' is A then ans will be AB' or even B is there in place of B' then AB so there is no point in the question of  writing A as A+0
 
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Given:

A'(A'+B'),A=A+0 apply it to the first term of A

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

=>(A'. 0') (A'+B') [DEMORGAN'S THEOREM]

=>(A'. 1) (A'+B') [(A'. 1)= A' IDENTITY]

=>A'(A'+B') [ASSOCIATIVE LAW]

=>(A'.A')+B' [A'.A'=A' IDEMPOTENT LAW]

=>A'+B' [(A.B)'=A'+B' DEMORGAN THEOREM]

=> (A.B)' [CORRECT ME WHERE I HAVE GONE WRONG BECAUSE MY ANSWER IS NOT GIVEN IN THE OPTION [LINK FOR BOOLEAN RULES=http://www.electronics-tutorials.ws/boolean/bool_6.html  ]

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A'(A'+B')
=A'A'+A'B'
=A'+A'B'
=A'
=(A+0)
=A
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Don't consider A = A+0 because in expression can't find A

so, 

This expression can be simplified as:
= A'A' + A'B'
= A' + A'B'
= A'(1 + B') // 1 + B' = 1
= A'
Answer:

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