Question 1 [Numbers and Set Theory] (a) (i) Suppose A, B, C are subsets of some set X. Draw a Venn diagram for theexpression (AB) U ((BNC) \ A). (ii) Find an expression for (A\B)u((BNC)\A) that uses only union, intersection,and complement. (iii) For the sets D = {x € Z | x² ≤ 40} and E = {x € Z | there exists y € Z such that 3y = x} write down D and D n E explicitly. (b) (i) Does Z6 satisfy the law of the multiplicative inverse? In other words, for each x € Z6 does there exist y € Z6 such that xy = 1 mod 6? Justify your answer.

x < - 0s <- 0

while (x < n)

X <- x + 1

s < s + 2*x - 1}

return s

Prove that s = x² is an invariant of the loop. (c) Java offers functionality for creating arrays of floating point numbers, which have the type float []. The length of such an array is specified as an int variable.

Consider the set of all possible arrays of type float []. Is the cardinality of this set finite, countable, or uncountable? Discuss the relationship of this set to the sets of lists and streams of numbers that we defined.