Download Test Paper - Discreet Mathematics for Computer Science - Gujarat Technological University - Masters in Computer Application - 1st Semester 2010 and more Study notes Computer Applications in PDF only on Docsity! 1 Seat No.: _____ Enrolment No.______ GUJARAT TECHNOLOGICAL UNIVERSITY MCA Sem-I Examination January 2010 Subject code: 610003 Subject ame: Discreet Mathematics for Computer Science Date: 21 / 01/ 2010 Time: 12.00 -2.30 pm Total Marks: 70 Instructions: 1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks. Q.1 (a) Define “Boolean expression”. Show that [a * (b’⊕ c)]’ * [b’⊕ (a * c’)’]’ = a * b * c’ 07 (b) Define “Symmetric Boolean expression”. Determine whether the following functions are symmetric or not: (i) a’bc’ + a’c’d + a’bcd + abc’d (ii) abc’ + ab’c + a’bc + ab’c’ + a’bc’ + a’b’c 07 Q.2 (a) Define “Universal quantifier” and “Existential quantifier”. (i) Express the following sentences into logical expression using First Order Predicate Logic: “All lines are fierce” “Some student in this class has got university rank” (ii) Show the following implication without constructing the truth tables first and thereafter show it through the truth tables. (P→Q) →Q => (P ∨ Q) 07 (b) Define equivalence relation. Let Z be the set of integers and R be the relation called “Congruence modulo 5” defined by R = {<x,y> | x )( yxZyZ −∧∈∧∈ is divisible by 5} Show that R is an equivalence relation. Determine the equivalence classes generated by the elements of Z. 07 OR (b) Define “compatibility relation” and “maximal compatibility block”. Let the compatibility relation on a set (x1, x2,…., x6} be given by the matrix x2 1 x3 1 1 x4 0 0 1 x5 0 0 1 1 x6 1 0 1 0 1 x1 x2 x3 x4 x5 Draw the graphs and find the maximal compatibility blocks of the relation. 07 2 Q.3 (a) Define “Composite relation” and “Converse of a relation”. Given the relation matrix MR of a relation R on the set {a, b, c}, find the relation matrices of ~R (Converse of a R), R 2 = R o R and R o ~R. MR = 1 0 1 1 1 0 1 1 1 07 (b) Prove the following Boolean Identities: (i) babaa ⊕=⊕⊕ )''( (ii) babaa *)''*(* = 04 (c) Find the six left cosets of H = {p1, p5, p6} in the group 〈 S3, * 〉 , given in the following table: * p1 p2 p3 p4 p5 p6 p1 p1 p2 p3 p4 p5 p6 p2 p2 p1 p5 p6 p3 p4 p3 p3 p6 p1 p5 p4 p2 p4 p4 p5 p6 p1 p2 p3 p5 p5 p4 p2 p3 p6 p1 p6 p6 p3 p4 p2 p1 p5 03 OR Q.3 (a) (i) Define “Partial order relation” and “Chain”. (ii) The following figure gives the Hesse diagram of a partially ordered set <P, R>, where P = {x1, x2, x3, x4, x5}. x1 x2 x3 x4 x5 Find which of the following are true: x1 R x2, x4 R x1, x1 R x1, and x2 R x5. Find the upper and lower bounds of {x2, x3, x4}, {x3, x4, x5}, {x1, x2, x3} 07 (b) Show that (i) aa =+ 0 (ii) '1 aa =+ (iii) 0=+ aa (iv) 1'=+ aa where '()'*( ababa ⊕=+ * b) 04 (c) Show that 〈 S3, * 〉 as given in the above table [i.e. Q.3(c) main part] is a group. [Note: Only one non-trivial example to show associativity will be sufficient. 03
