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I will notate & for "and" and v for "or"

A: I got X
B: I went to school.

~(A v B) = Neither I got X nor did I go to school.

which is equivalent to ~A & ~B = I did not get X and I did not go to school.

But what I have trouble with is

~A v ~B is I did not get X or I did not go to school.

which is equivalent to ~(A & B), but I do not know how to translate ~(A & B) into an English sentence.

Can anyone help me?

Also, what is the difference between Both I love 1 and I love 2 and I love both 1 and 2?

Are both sentences equal to A & B ?

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    & and ^ have the same meaning as conjunction in all notations I know. v means disjunction. I've edited for that purpose. – virmaior Sep 24 '14 at 8:58
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~(A & B) is It is not the case that I both got X and went to school. You would rarely hear a sentence like this in natural English, which is why it is hard to translate.

Both of your sentences could be expressed by A & B in a simple logic such as the one you're using. If you were working with a predicate logic, the first would be L(A) & L(B), the second would be L(A & B).

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