$1)$ According to the Put$-$Call Parity, an American put option price is determined by an American call optionprice, the stock price, and the risk$-$free interest rate.

True$/$False

$2)$ It may be optimal to early exercise an American call option, while it is never optimal to exercise an American put option early.

True$/$False

$3)$ The lower bound of the American put option is Max $($PV$($X$)-$S$0,0).$

True$/$False