1. Find an integer part of the binomial expansion of (√√2+ √3)5 (n-th root of prime number is always irrational). 2. Using the mathematical induction, prove n Σ(k² + 2k - 2) = n(2n² + 9n – 5), k=1 3. Simplify the given composite set, using formulas, and then check the result, using Venn diagrams. ((MUN) U (MAN)) M 4. Check if the following relations are partial order, or equivalence, or none. For the equivalence relations, find equivalence classes and par- tition, for partial orders, draw a Hasse diagram. (i) A= {a, b, c, d, e, f}, RA= {[a, a], [b, b], [c, c], [d, d], [e, e], [ƒ, ƒ], [c, d], [f, c], [d, f], [c, ƒ], [d, c], [f,d]}, (ii) B= {a, b, c, d, e, ƒ}, RB = {[a, a], [b, b], [c, c], [d, d], [e, e], [f, ƒ], [c, a], [b, a], [b, ƒ], [c, d], [c, b], [a, d], [f, d], [c, f]}. 5. For the given maps, check if they are functions, injective, surjective, bijective. In case the function is bijective, find an inverse mapping. (i) f: AA, A = {a, b, c, d, e}, f = {[b, a], [a, b], [d, a], [c, d], [d, e], }, (ii) g: BB, B = {a, b, c, d, e, f}, g(b) = b, g(a) = a, g(f) = a, g(c) = d, g(d) = e, g(e) = f, (iii) h: CD, C = {1, 2, 3, 4, 5, 6}, D = {a, b, c, d, e, f}, h(1) = b, h(2) = c, h(3) = f, h(5) = d, h(4) = e, h(6) = a, (iv) 1: R→ [2,0], h(x) = 3x² + 2, (v) k: RR, 1(x) = 3x+4. : 6. Let f RR, h R R and 9: RR, where f(x) = 3x + 2, 9(x) = 31 and h(x) = (sin(x))2. Find the composition fogoh. 7. Let A = {a, b, c, d, e}, B = {1, 2, 3, 4, 5, 6} and f: A → B is given by f(a) = 2, f(b) = 1, f(c) = 3, f(d) = 5, f(e) = 6 and 9: BA is given by g(2) = a, g(1) = b, g(3) = c, g(5) = d, g(6) = e, g(4) = c. Prove if, or if not gf-1, that is, if g is, or it is not the inverse function of f. = 8. There are 9 different Latvian and 5 different English books. By how many ways can we can create a package of 4 books which contains at least 3 Latvian books?

Hi, this is test for preparation (not a graded assignment) for the exam from discrete math course that consists of 8 questions, i would like to recieve a full explanation to all of the questions. If asking a solution to 8 questions at one time is against the rules of this platform, i would ask you to do the question number 7. Thanks in advance