Discrete Mathematics — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

a) Construct the truth table for the logical statement: b) Let A= Z + (the set of positive integers). Define a relation R onA by: a R b if and only if |a -b| S 2` Determine whether the relation R is: a) Define a partially ordered set (POSET). Show that (P({a, b, c},E) partially ordered sets. b) Showthat @({1, 2, 3},E) is a lattice. c) Define the maximum element of a set. Find god( 1800, 756). a) b) Determine whether or not each of the following integers is a prime Use the fundamental theorem of Arithmetic to prove that for no natural` number n does the integer 6n terminate in o. 6 Prove by mathematical induction that 1 + 2 + 3 + .... + 77 = for any natural number n. a) b) c) Solve the recurrence relation c}„ = 3cr„_I, # 21 given ao = 1. How many integers between I and 500 are : (ii) divisible by 3 but not 5 or 6? There are 15 questions on a multiple choice exam and five possible\ answers to each question. (i) In how many ways can be exam by answered? (ii) In how many ways can the exam be ansvyered with exactly 8 answers correct? a) Ifz, and A4are Lattices, then show that fxM' is also a lattice. Then determine if g j.a z.. j o in homomorphism, I.z.. meet homomorphism, 7.7.I.. homomoxphism r.v. order homomoxphism c) Define Lattice as analgebra. Let f be the set of divisors of30 equ`ipped with divisibility order. Find the complement of each member off,. 6 a) Put the function /rxj, :*2, x3J = i-/ A (*2 V :r3 ) in the conjunctive normal form. b) Find the minimal form of f = X'yz + Jr'yz' + Jr'y'z + jry'z' + jry'z c) Determine the symbolic representation of the circuit p = ('.¥{ + ]c2)'L*.i 6.5