Tiêu đề 3140708 - DM 2021W.pdf ✅

1 Seat No.: ________ Enrolment No.___________ GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–IV (NEW) EXAMINATION – WINTER 2021 Subject Code:3140708 Date:24/12/2021 Subject Name:Discrete Mathematics Time:10:30 AM TO 01:00 PM Total Marks: 70 Instructions: 1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks. 4. Simple and non-programmable scientific calculators are allowed. Q.1 (a) Show that for any two sets A and B, A − (A ∩ B) = A − B. 03 (b) If S = {a, b, c}, find nonempty disjoint sets A1 and A2 such that A1 ∪ A2 = S. Find the other solutions to this problem. 04 (c) Using truth table state whether each of the following implication is tautology. a) (p⋀r) → p b) (p⋀q) → (p → q) c) p → (p⋁q) 07 Q-2 (a) Given S = {1, 2, 3, − − − − ,10} and a relation R on S. Where, R = {〈x, y〉|x + y = 10} . What are the properties of relation R? 03 (b) Let L denotes the relation “less than or equal to” and D denotes the relation “divides”. Where xDy means “x divides y”. Both L and D are defined on the set {1, 2, 3, 6}. Write L and D as sets, find L ∩ D. 04 (c) Let X = {1, 2, 3, 4, 5, 6, 7} and R = {〈x, y〉|x − y is divisible by 3}. Show that R is an equivalence relation on. Draw the graph of R. 07 OR (b) Define equivalence class generated by an element x ∈ X. Let Z be the set of integers and let R be the relation called “congruence modulo 3” defined by R = {〈x, y〉|x ∈ Z⋀y ∈ Z⋀(x − y) is divisible by 3} Determine the equivalences classes generated by the element of Z. 07 Q.3 (a) Let f(x) be any real valued function. Show that g(x) = f(x)+f(−x) 2 is always an even function where as h(x) = f(x)−f(−x) 2 is always an odd function. 03 (b) The Indian cricket team consist of 16 players. It includes 2 wicket keepers and 5 bowlers. In how many ways can cricket eleven be selected if we have select 1 wicket keeper and at least 4 bowlers? 04 (c) Let A be the set of factors of particular positive integer m and ≤ be the relation divides, that is ≤= {〈x, y〉|x ∈ A⋀y ∈A⋀(x divides y)} Draw the Hasse diagrams for a) m = 45 b) m = 210. 07 OR Q-3 (a) Find the composition of two functions f(x) = e x and g(x) = x 3 , (f ∘ g)(x) and (g ∘ f)(x). Hence, show that (f ∘ g)(x) ≠ (g ∘ f)(x). 07 (b) In a box, there are 5 black pens, 3 white pens and 4 red pens. In how many ways can 2 black pens, 2 white pens and 2 red pens can be chosen? 04 (c) Let A be a given finite set and ρ(A) its power set. Let ⊆ be the inclusion relation on the elements of ρ(A). Draw Hass diagram for 〈ρ(A), ⊆〉 for a) A = {a, b, c} b) A = {a, b, c, d} 07