If P= 5 m: m N and Q= 5^m: m N , where N is set of natural numbers, then

Step-by-step solution

The set P is defined as P = 5m : m N = 5, 10, 15, 20, 25, 30, . The set Q is defined as Q = 5^m : m N = 5^1, 5^2, 5^3, = 5, 25, 125, 625, . Comparing the elements of P and Q , we observe that every element of Q is of the form 5^m . Since m N , 5^m = 5 5^ m-1 . Let k = 5^ m-1 . Since m 1 , k is a natural number. Thus, 5^m = 5k , which is an element of P . Therefore, every element of Q is also an element of P , which implies Q P . Checking other options, P contains elements like 10 which are not in Q , so P Q and P Q