We are throwing two dice so,

Sample space S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Sample space contains 36 elements,

A= Number four appears on first die

A={(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)}

B=Number 5 on second die

B={(1,5), (2,5), (3,5),(4,5),(5,5),(6,5)}

P(A∩B)={(4,5)}

Therefore A and B are independent events