Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)

Question

Prove that the following two languages are undecidable.

1. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq  \emptyset\}$.
2. $PREFIX-FREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefix-free}\}$.