$\mathrm{}$Joe has been having a few good days. Recently, Joe went to a flea market and purchased an old piano for $$100.$ To his surprise, Joe learned that the piano has a fair market value of $$15,000,$ despite having paid only $$100$ for it$.$ If that weren$$t enough, when Joe opened the lid of the piano to look at the strings, he discovered $$25,000$ of cash. 

 

a$.$ Does Joe have to income due to the appraisal at $$15,000$ when compared to his purchase price? Why or why not? Please explain.

b$.$ Does Joe have $$25,000$ of taxable income due to his discovery of cash in the piano? Why or why not? Please explain.