How to Solve The Redundant Literal Rule (AND)?

Question

(A . B = A . (A̅ +B))

How to prove them? (Tried so many times)

Answer 1

$$A\land (\lnot A \lor B)=(A\land\lnot A)\lor (A\land B)=0\lor (A\land B)=(A\land B)$$

Answer 2

A . (A̅ +B)

= A.A̅ + A.B

But A.A̅ = 0 ( Inverse or Complement Law)

= A.B

Answer 3

Hint: Use the fact that the product of a variable and its complement is $0$. Thus, $AA' =0$. Can you take it from here? Hope it helps.

Answer 4

Use the distributive law and you get

$$ A\cdot\bar{A} + A\cdot B. $$

$\text{True AND False}$ results in $\text{False}$. Which give $A\cdot \bar{A} = 0$. Again, $\text{False OR X}$ always results in $X$, hence we get $0 + AB = AB$. Thus,

$$ A(\bar{A} + B) = A\cdot\bar{A} + AB = AB$$

Answer 5

A•(Ā + B) = A•Ā + A•B using the distributive property.

EDIT: This is in response to the OP's request to explain further. Wanted to add this as a comment, but encountered an error.