Other ways of saying
Ich esse weder Äpfel noch Birnen.
appear to be
Ich esse keine Äpfel und Birnen.
Ich esse keine Äpfel oder Birnen.
Are both sentences correct? Is one more unusual than the other?
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All three sentences are totally correct in German language. I don't think there is any preference for one or the other.
Despite of mathematics language very often does not differentiate between the logical "and" and the logical "or".
I cannot even find a slight different meaning in the two sentences.
But this is nothing specific to German language. Just translate the three sentences:
I neither like apples nor pears
I don't like apples and pears
I don't like apples or pears
All sound quite correct in English language as well.