In the proof that |(0, 1)| > |N|, we use Cantor's Diagonal Method, where we change the nth digit dnn of the nth number r, E (0, 1) in a purported fixed list of all numbers r E (0, 1) by changing dnn to dn = 1 if dnn + 1 and letting dn = 2 otherwise. Then we form a number r* 0.d,d2 ... and claim that because r* is not in the purported list, therefore |(0, 1)| > |N]. State the function used in the proof clearly and, using the definition of two sets being equal in cardinality |A| = |B|, explain why the proof shows that the cardinality of (0, 1) is strictly bigger than the cardinality of N.

Define the Fibonacci numbers as Fo= 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1 − F2 = (-1)" for n ≥ 1 and use this to deduce that gcd(Fn, Fn-1) = 1 for all n ≥ 1.

(b) Let J₁ 100] 0100 001 1000 and J₂ = Explain whether J₁ and J₂ are similar. t000] 010 0011 000 1

Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.