Let F = {0, 1} be the field with two elements, and let · denote the standard dot product in F100, defined by (x1, ... , x100) · (yı,... , y100) = x1yı + … + x100Y100 € F. Denote by Cj, ... , CM the clubs that the town has formed under the new rules. (So that M is the number of distinct clubs.) After numbering the inhabitants from 1 to 100, associate to each C; a vector v; in F100, whose i-th entry is equal to 1 precisely if the i-th inhabitant belongs to C;. (a) Show that the vectors V1, . , UM satisfy if i + j, V¡Vj = if i = j, (b) Use the result of (a) to show that the vectors (v1, .. , UM) are linearly independent in F100. (c) Conclude that M < 100.

Let à = ст -5' -4 Find ||a, the norm of a.

(b) Prove that Z/4Z is not a field.

Let B denote a Boolean algebra. Prove the identity V a, b e B, (a · b = 0) ^ (a + b = 1) = a = b. That is, prove the complement b is the unique element of B which satisfies (b · b = 0)^ (b + b = 1).