MATH 13 Study Guide - Final Guide: Venn Diagram, Maxima And Minima, Joule
Document Summary
Total marks = 100 (per question in brackets) Unless otherwise stated, include all your working for full credit. Try all parts of every question, even if you can"t do the rst part! Prove your assertion. (6) (d) suppose that x, y and z are sets with cardinalities |x| = 4, |y| = 2 and |z| = 3. There are three possible cardinalities for the set (x y) p (z) Dec2. (a) complete the table of remainders for x2 modulo 4: (4) x mod 4 x2 mod 4. 0 1 2 3 (b) suppose that n is even but not divisible by 4. Use part (a) to prove that there are no integers (6) x, y such that n = x2 + 3y2. 1 (n + 1)2 < 2 1 n + 1 (b) prove by induction that. 1 k2 2 1 n (5) (5) Then a + b := |a b|
