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Lepton

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Computer
1. Complement of a Boolean input variable in a circuit?

This question is an example in a pdf I have. Yes I also think that the D must be inverted to /D. Now I noticed that A is inverted 2 times so it remains same. And thanks for helping once again.
2. Complement of a Boolean input variable in a circuit?

I am having trouble understanding that how this complement of input variable is working in the circuit below. Please help. The Process:-- Our motive is to draw a circuit diagram with only a NOR gate of the Boolean expression :-- Step 1:-- Step 2 :-- Step 3 :-- Now from step 2 to step 3 we have to remove the inverters and complement the input variables as marked in the image. So, my question is how this complementation is working? In some places it remains the same (in below 'D' & 'A') and in some places its complements. Any help will be appreciated.
3. How to minimize this Boolean expression in shorter form?

@KipIngram Thanks
4. How to minimize this Boolean expression in shorter form?

For example I have simplified this equation. 1st step arranging the literals. 2nd step applying Distributive Law 3rd step summation of y and its complement is equals to 1. So, like this how can I solve those above equations. And yes I am familiar with Karnaugh map.
5. How to minimize this Boolean expression in shorter form?

How can I minimize these two Boolean expressions in shorter form. I have tried quite hard to solve but unable to do so. I can't apply any of the basic postulates here. Please help me.
6. Boolean Algebra proof?

Now my doubt is cleared. Thanks for helping.
7. Boolean Algebra proof?

Hello all, This is my first post here as I am new here. My question is in this proof listed below in the image. I can't understand a line how is this possible in Boolean Algebra. Any help will be appreciated.