Download PTU BCA 2020 Dec 1st Sem 10045 Mathematics I Question Paper

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Total No. of Pages : 03
Total No. of Questions : 16

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains SIX questions carrying TEN marks each and students have to attempt any FOUR questions.

SECTION-A

Write briefly :

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1. If A, B, C are any sets, prove that A — (B?C) = (A-B)n(A-C).
2. Define partition of sets.
3. Let X={1,2,3,4,5,6,7,8,9, 10}. The family '{{1,4,8}, {3,5,9},{2, 7}, {6, 10}} is a partition of X. Determine the equivalence relation corresponding to the above partition.
4. Let X={1,2,3,4} and R = {(x, y): x > y}: Draw the diagraph and matrix of R.
5. Using truth table, prove that ~ (p?q) = p?~q.

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6. Given the proposition over the natural numbers p: n<4, q :2n> 17 and r : n is a divisor of 18. What are the truth sets of p?q and q ? r.
7. Prove that the number of edges in a complete graph with n vertices is n(n-1)/2.
8. Draw a simple planar graph with 6 nodes and 11 edges.
9. Define recurrence relation with example.
10. Solve the recurrence relation S(K) - S(K - 1)-S(K-2)=0.

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SECTION-B

1. a) State and prove De Morgan’s Laws for sets.
   b) The relation R is defined by (a, b) ? R if and only if 5 divides b — a. Show that R is an equivalence relation.
2. a) Let R = {(a, b): |a —b| =1} and S = {(a, b): a — b is even} are two relations on A={1,2,3,4}. Then
   

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   i) Find matrices of R and S.
   ii) Draw diagraphs of R and S
   iii) Using matrices of R and S, find the relation RS.
   b) Test the validity of “If my brother stands first in the class, I will give him a watch. Either he stood first or I was out of station. I did not give my brother a watch this time. Therefore I was out of station.”
3. a) Over the universe of Books, define the proposition B(x): x has a blue cover, M(x): x is a mathematics book, U(x): x is published in United Estate and R(x, y): The bibliography of x includes y.
   

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   Translate into words :
   i) (?x)(M (x)? ~ B(x)).
   ii) (?x)(M (x) ?U(x) ? B(x))
   iii) (?x)(~ B(x))
   Express using quantifiers :
   

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   i) Every book with blue cover is a mathematics book.
   ii) There are mathematics books that are published outside the United States.
   iii) Not all books have bibliography.
   b) Use Mathematical Induction to show that 1+2 +4 + ... +2ⁿ=2ⁿ⁺¹-1.
4. Using Dijkstra’s Algorithm, find shortest path from A to D.
   

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