CISC102 Winter 2024Quiz 4: Relations Reminders◦ Answer each question to the best of your abilities, and show all of your work◦ You may use your notes from class but are prohibited from using external resources.◦ You must handwrite your solutions. Electronic solutions will automatically receive a 0 Insufficient justification will result in a loss of marks.Questions1. (2 marks) Describe in words the following relation, where n is some fixed integer:R = {(a,b) ∈ Z2 | n|(a − b)}2. (5 marks) Given the recurrence relation an = f · an − 1 + ℓ with a0 = 2. Let f be the integer corresponding to the first letter of you First name and let ℓ be the integer corresponding to the first letter of your Last name (eg. Selena Gomez would have f = 19 ℓ = 7 since S is the 19th letter of the alphabet and G is the 7th letter.. Write out the first 5 terms of the sequence.. Using the repeated substitution method from class, determine a closed form for the recurrence relation.. Check that your closed form holds for the first 5 terms. 3. (6 marks) Let R be the relation dai 写CISC102 Winter 2024 Quiz 4: RelationsC/C++ on the set of all bitstrings of length n, where aRb whenever a and b ‘correspond’ at the second position. That is, the second item for a,b is either both 0 or both 1.(a) Prove that R is an equivalence relation(b) Determine the equivalence classes of R.4. (5 marks) Complete the following induction proof by filling in each of the missing areas.Prove the closed form of this sumation:Proof.Base Case: Let n = 0, then the LHS gives The RHS is . So the base case holds.Induction Hypothesis: [Fill This Out]Inductive Step:ConsiderTherefore, we have proved by induction that .5. (7 marks) Let f : R → R be a function such that(a) Prove that f is a bijection.(b) Find f− 1(c) Find the image of the set S = {−3, 0.5, 5/2, π} under f. That is, find imagef (S).6. (8 marks) Given the recurrence relation an = 8an−1 −12an −2 with a0 = −2 and a1 = 0. Prove using strong induction that the closed form isan = 6n − 3 · 2n WX：codehelp
