Disagreements with affirmative sentences are made with ‘no + pronoun + auxiliary + n’t/not’.

He is drunk. No, he isn’t.
You are joking. No, I am not.
I think she knows him. No, she doesn’t.

Note that but is often used instead of no in disagreements with a question or an assumption.

Why did you steal my purse? But, I didn’t. (Here we are disagreeing with the question.)

Disagreements with negative sentences are made with ‘yes + pronoun + auxiliary’.

You can’t understand it. Yes, I can.

Addition to remarks

Affirmative additions to affirmative remarks are made with ‘so + auxiliary + subject’.

He likes fish. So do I.
She must work hard. So must I.
He was late for the meeting. So was I.
He has finished his homework. So have I.

Negative additions to negative remarks are made with ‘nor/neither + auxiliary + subject’.

John doesn’t like sweets. Neither do I.
She didn’t believe it. Neither did I.
I couldn’t solve the problem. Neither could my mother.

Negative additions to affirmative remarks are made with ‘but + subject + auxiliary + n’t/not’.

Compare:

He speaks English. But I don’t.
He doesn’t speak English. Nor do I.
I found the answer. But Alice didn’t.
I couldn’t find the answer. Neither did Alice.
He can cook. But his wife can’t.
He can’t cook. Nor can his wife.
I can play chess. But my sister can’t.
I can’t play chess. Neither can my sister.

Affirmative additions to negative remarks are made with ‘but + subject + auxiliary’.

He doesn’t know how to cook. But I do.
I didn’t see the film. But my sister did.
He can’t play chess. But I can.
She wasn’t late. But I was.