Consider the following sets of points in the complex plane A = ( 2n 5 ) + i ( 2n 5 ) : n Z and B = ( 2n 5 ) + i ( 2n 5 ) : n Z . Which of the following statements is TRUE?

Step-by-step solution

The set A consists of complex numbers of the form z_n = ( 2n 5 ) + i ( 2n 5 ) = e^ i 2n 5 for n Z . Since e^ i 2(n+5) 5 = e^ i 2n 5 e^ i 2 = e^ i 2n 5 , the values repeat after every 5 integers. Thus, A contains exactly 5 distinct elements, making A a finite set. The set B consists of complex numbers of the form w_n = ( 2n 5 ) + i ( 2n 5 ) = e^ i 2n 5 for n Z . If two elements in B are equal, then e^ i 2n₁ 5 = e^ i 2n₂ 5 for some n₁, n₂ Z . This implies 2n₁ 5 - 2n₂ 5 = 2k for some k Z . n₁ - n₂ = 5k Since n₁, n₂, k