Define sets A and B as follows A n Z n 8r 3 for some inte

Define sets A and B as follows: A = {n Z | n = 8r -3 for some integer r} and B = {m Z | m = 4s + 1 for some integer s}.
Is A B?
Is B A?
Justify your answers by providing a proof if the statement is true or a disproof if the statement is false.

Solution

8r-3 = 4s+1

8r - 4s = 4

2r - s = 1

r = (1 + s)/2 or s = 2r - 1

For s= even integer, r is not an integer.

r such not exist for all s. So, A is a subset of B.