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obviously its a piano sonata. the only instrument is the piano
Yes, it was composed for the piano by Beethoven. It's official title is Piano Sonata No. 8 in C minor.
A Piano Sonata
It is a Sonata For Violin & Piano.
The name is given in the title (although the name was given after Beethoven died).
Piano Sonata no. 14 in C# minor - "Moonlight Sonata".