Solapur University Question Paper
M.C.A. – I (Semester – I) Examination, 2014
(Commerce and Management Faculty)
DISCRETE MATHEMATICS (Old)
Day and Date : Tuesday, 13-5-2014 Max. Marks : 70
Time : 11.00 a.m. to 2.00 p.m.
Instructions : 1)Q. No. 1 and 7 are compulsory.
2) Solve any two questions from Q. No. 2, 3 and 4.
Solve any one question from Q.No. 5 and 6.
3) Figures to the right indicate full marks to qeustion.
1. A) Choose the correct alternative. 10
1) The equivalence of two proposition A and B is denoted by
i) A ⇔ B ii) A → B iii) A ≡ B iv) Both i and iii
2) p → q = _______
i)⎤p ∨ q ii) ⎤ p ∨⎤ q iii) p ∨ q iv) q ∨ p
3) A graph in which all vertices set can be partitioned into two set is and for
every edge both vertices are from distinct set is called
i) Simple graph ii) Regular graph
iii) Bipartite graph iv) Planer graph
4) A Graph without loop and parallel edges is called _______ graph.
i) Simple graph ii) Regular Graph
iii) Bipartite graph iv) Planer graph
5) If a* e = e* a = a then e is called
i) Inverse of a ii) Inverse of b
iii) Identity element iv) Factor of a
Seat
No.
SLR-GL – 5 -2-
6) A partially ordered set in which every pair of element has both Least
Upper bound and Greatest lower bound is called
i) Inverse set ii) Lattice
iii) Complement set iv) Inverser set
7) _______ set contains collection of all subsets of given set.
i) union ii) power iii) partition iv) empty
8) Set B {0, 1} is a group with _______ operation.
i) mul mod 2 ii) addition iii) add mod 2 iv) mod 2
9) A formula which consist of product of elementary sum is called
i) CNF ii) DNF iii) Minterms iv) Maxterms
10) A formula which consist of sum of elementary product is called
i) CNF ii) DNF iii) Minterms iv) Maxterms
B) True or False ; 4
1) Prefix and postfix are called polish notations.
2) Relation is always a function.
3) If a is generator inverse of a is also generator.
4) If all vertices having degree odd graph has Euler circuit.
2. A) Prove that premises p → q, q →r, s →~ r and q∧ s are inconsistent. 7
B) Prove without using truth table. 7
¬p ↔ q ≡ p ↔ ¬q
-3- SLR-GL – 5
3. A) If S = {1, 2, 3, 4, 5} and if function f, g, h : S → S are given by 7
f = {(1, 2) (2, 1), (3, 4), (4, 5), (5, 3)}, g = {(1, 3), (2, 5), (3, 1), (4, 2), (5, 4) },
h = {(1, 2), (2, 2), (3, 4), (4, 3), (5, 1)}.
i) Find f o g and g o f ii) Find f–1 and g–1
iii) Show (f o g)–1 = (g–1 o f–1) iv) Show ( f o g)–1 ≠ (f–1 o g–1)
B) If A = {1,2, 3, 4} and the R is relation on set A is defined by (a, b) R (c, d) if
a + b = c + d verify that A is equivalence relation ? Find the Quotient set of A
by R i.e. (A/R). 7
4. A) Let A be the set of +ve factors of 45 and let ≤ be the relation divides
i.e. ≤ = {<x, y>|x, y ∈ A and x divides y}. Draw the Hasse diagram. 7
B) Find the value of prefix expression + – *
235/ ↑ 234. 7
5. A) Define Graph. Explain Euler Graph and Hamilton Graph in detail with
examples. 7
B) Define functions. Explain types of function. 7
6. Find codewords generated by parity check matrix. 14
0 1 1 1 0 0 1
1 1 1 0 0 1 0
1 0 1 1 1 0 1
7. Explain cyclic group. Show that the set of {1, –1, i, – i} is a cyclic group under
multiplication. 14
__________________
M.C.A. – I (Semester – I) Examination, 2014
(Commerce and Management Faculty)
DISCRETE MATHEMATICS (Old)
Day and Date : Tuesday, 13-5-2014 Max. Marks : 70
Time : 11.00 a.m. to 2.00 p.m.
Instructions : 1)Q. No. 1 and 7 are compulsory.
2) Solve any two questions from Q. No. 2, 3 and 4.
Solve any one question from Q.No. 5 and 6.
3) Figures to the right indicate full marks to qeustion.
1. A) Choose the correct alternative. 10
1) The equivalence of two proposition A and B is denoted by
i) A ⇔ B ii) A → B iii) A ≡ B iv) Both i and iii
2) p → q = _______
i)⎤p ∨ q ii) ⎤ p ∨⎤ q iii) p ∨ q iv) q ∨ p
3) A graph in which all vertices set can be partitioned into two set is and for
every edge both vertices are from distinct set is called
i) Simple graph ii) Regular graph
iii) Bipartite graph iv) Planer graph
4) A Graph without loop and parallel edges is called _______ graph.
i) Simple graph ii) Regular Graph
iii) Bipartite graph iv) Planer graph
5) If a* e = e* a = a then e is called
i) Inverse of a ii) Inverse of b
iii) Identity element iv) Factor of a
Seat
No.
SLR-GL – 5 -2-
6) A partially ordered set in which every pair of element has both Least
Upper bound and Greatest lower bound is called
i) Inverse set ii) Lattice
iii) Complement set iv) Inverser set
7) _______ set contains collection of all subsets of given set.
i) union ii) power iii) partition iv) empty
8) Set B {0, 1} is a group with _______ operation.
i) mul mod 2 ii) addition iii) add mod 2 iv) mod 2
9) A formula which consist of product of elementary sum is called
i) CNF ii) DNF iii) Minterms iv) Maxterms
10) A formula which consist of sum of elementary product is called
i) CNF ii) DNF iii) Minterms iv) Maxterms
B) True or False ; 4
1) Prefix and postfix are called polish notations.
2) Relation is always a function.
3) If a is generator inverse of a is also generator.
4) If all vertices having degree odd graph has Euler circuit.
2. A) Prove that premises p → q, q →r, s →~ r and q∧ s are inconsistent. 7
B) Prove without using truth table. 7
¬p ↔ q ≡ p ↔ ¬q
-3- SLR-GL – 5
3. A) If S = {1, 2, 3, 4, 5} and if function f, g, h : S → S are given by 7
f = {(1, 2) (2, 1), (3, 4), (4, 5), (5, 3)}, g = {(1, 3), (2, 5), (3, 1), (4, 2), (5, 4) },
h = {(1, 2), (2, 2), (3, 4), (4, 3), (5, 1)}.
i) Find f o g and g o f ii) Find f–1 and g–1
iii) Show (f o g)–1 = (g–1 o f–1) iv) Show ( f o g)–1 ≠ (f–1 o g–1)
B) If A = {1,2, 3, 4} and the R is relation on set A is defined by (a, b) R (c, d) if
a + b = c + d verify that A is equivalence relation ? Find the Quotient set of A
by R i.e. (A/R). 7
4. A) Let A be the set of +ve factors of 45 and let ≤ be the relation divides
i.e. ≤ = {<x, y>|x, y ∈ A and x divides y}. Draw the Hasse diagram. 7
B) Find the value of prefix expression + – *
235/ ↑ 234. 7
5. A) Define Graph. Explain Euler Graph and Hamilton Graph in detail with
examples. 7
B) Define functions. Explain types of function. 7
6. Find codewords generated by parity check matrix. 14
0 1 1 1 0 0 1
1 1 1 0 0 1 0
1 0 1 1 1 0 1
7. Explain cyclic group. Show that the set of {1, –1, i, – i} is a cyclic group under
multiplication. 14
__________________
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