VIGNAN’S FOUNDATIONS FOR SCIENCE, TECHNOLOGY & RESEARCH
SCHOOL OF APPLIED SCIENCES & HUMANITIES
DEPARTMENT OF MATHEMATICS & STATISTICS
DR. R. PANDURANGA RAO
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Lagrange’s theorem
Group
Rings
Field
Tautology
Contradiction
Equivalence
Normal form CNF and DNF
K-Maps
Permutations and combinations
Recurrence relations and generating functions
Planner and Bipartite graphs.
Matrix representation of graphs
Graph Collaring and chromatic number
Isomorphism
BFS and DSF
Kruskal's algorithm (MST)
Prim’s algorithm (MST)
S. No
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Topic
Roll Number/ Section
242FA04001/CSE-A
Lagrange’s theorem
242FA04008/CSE-A
Groups
242FA04040/CSE-A
Rings
242FA04034/CSE-A
Field
242FA04048/CSE-A
Tautology
242FA04217/CSE-D
Contradiction
242FA04205/CSE-D
Equivalence
242FA04223/CSE-D
Normal form CNF and DNF
242FA04060/CSE-A
K-Maps
242FA04224/CSE-D
Permutations and combinations
Recurrence relations and generating 242FA04085/CSE-B
functions
242FA04249/CSE-D
Planner and Bipartite graphs.
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13
14
15
16
17
18
Matrix representation of graphs
Graph Collaring and chromatic number
Isomorphism
BFS and DSF
Kruskal's algorithm (MST)
Prim’s algorithm (MST)
242FA04234/CSE-D
242FA04096/CSE-B
242FA04100/CSE-B
242FA04097/CSE-B
242FA04068/CSE-B
242FA04119/CSE-B
1. Lagrange’s theorem (notes)
1. Define Lagrange’s theorem.
2. Let G be a group of order 20. What are the possible orders of subgroups of G? Justify
using Lagrange’s Theorem.
3. Let G be a group of order 21. Show that the order of any element a ∈ G must divide
21.
4. Let G = Z 7∗ = {1,2,3,4,5,6}, a group under multiplication modulo 7. Let H =
{1,2,4,5}. Use Lagrange’s Theorem to determine whether H could be a subgroup.
2. Group
1. Let G be a set of non-zero real numbers and a*b = ab/2. Show that (G,*) is an abelian
group.
2. Show that the set G = {1,-1,i,-i} is a group under multiplication.
3. Show that the set G = {1,ω,ω2} is a group under multiplication.
3. Rings
1. Define Ring.
2. Show that ℤ₅[x] (polynomials over ℤ₅) is a ring.
3. Prove that the set of all odd integers is not a ring under usual addition and
multiplication.
4. If R is a ring and a, b ∈ R, prove that a (b - c) = ab - ac.
4. Field
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3.
4.
Define Field.
Prove that the set of all real numbers forms a field.
Prove that the set of all rational numbers forms a field.
What is the condition for ℤₙ to be a field? Prove it.
5. Tautology
1. Define a tautology.
2. Show that p ∨ ¬p is a tautology. (pg. no. 27 – E-book)
3. Show that (p ∧ q) → (p ∨ q) is a tautology. (pg. no. 32 – E-book)
4. Show that [(p → q) ∧ (q → r)] → (p → r) is a tautology by using truth tables. (pg. no.
38, 12b
– E-book)
6. Contradiction
1. Define a Contradiction.
2. Show that p ∧ ¬p is a contradiction. (pg. no. 26 – E-book)
3. Check if 𝑝 ↔ ¬𝑝 is a contradiction.
4. Show that (𝑝 → 𝑞) ∧ (𝑝 ∧ ¬𝑞) is a contradiction.
7. Equivalence
1. Define Equivalence.
2. Show that ¬(p ∨ q) and ¬p ∧ ¬q are logically equivalent. (pg. no. 27Ebook)
3. Show that p → q and ¬p ∨ q are logically equivalent. (pg. no. 28 E-book)
4. Use truth tables to verify the commutative law p ∨ q ≡ q ∨ p. (pg.no. 38,
3a. E-book)
8. Normal form CNF and DNF
1. Define CNF and DNF.
2. Find the product-of-sums expansion of each of the Boolean function
F(x, y, z) = (x + z)y. (pg. no. 858 E-book)
3. Find the sum-of-products expansion for the function F(x, y, z) = (x +
y)z. (pg. no. 856 – E-book)
4. Find the sum-of-products expansions of the Boolean function F(x, y) = x ¬y. (pg. no.
858 E-book)
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2.
3.
4.
9. K-Maps
Define K-map and how it is extracted from a truth table.
Find the K-maps for x y + ¬x y. (pg. no. 866 E-book)
Find the K-maps for ¬x y + x ¬y + ¬x ¬ y. (pg. no. 866 E-book)
Draw a K-map for a function in two variables and put a 1 in the cell representing x y.
(pg. no. 877 E-book)
10. Permutations and combinations
1. Define Permutations and combinations.
2. Suppose that there are eight runners in a race. The winner receives a gold medal, the
second-place finisher receives a silver medal, and the third-place finisher gets a
bronze medal. How many different ways are there to award these medals, if all
possible outcomes of the race can occur and there are no ties? (pg. no. 430 E-book)
3. How many ways are there to select five players from a 10-member tennis team to
make a trip to a match at another school? (pg. no. 434 E-book)
4. What is the expansion of (x + y) 4? (pg. no. 438, E-book)
11. Recurrence relations and generating functions
1. Define Recurrence relations and generating functions.
2. Find a recurrence relation and give initial conditions for the number of bit strings of
length n that do not have two consecutive 0s. How many such bit strings are there of
length five? (pg. no. 531 E-book)
3. Find the number of solutions of e1 + e2 + e3 = 17, where e1, e2, and e3 are
nonnegative integers with 2 ≤ e1 ≤ 5, 3 ≤ e2 ≤ 6, and 4 ≤ e3 ≤ 7. (pg. no. 569 E-book)
4. Find a recurrence relation for the number of ways to climb n stairs if the person
climbing the stairs can take one stair or two stairs at a time. (pg. no. 537, 11a E-book)
12 Planner and Bipartite graphs.
1. Define Planner and Bipartite graphs.
2. Are the graphs G and H displayed bipartite? (pg. no. 690 E-book)
3. Determine whether the given graph is planar. If so, draw it so that no edges cross. (pg.
no. 760 E-book)
4. Is K5 nonplanar? (pg. no. 761, 11 E-book)
13. Matrix representation of graphs
1. What is the Matrix representation of graphs called? Give an example.
2. Use an adjacency matrix to represent the graph shown. (pg. no.704 E-book)
3. Find the in-degree, out-degree and adjacency matrix of the graph shown. (pg. no.711
E-book)
4. Draw a graph with the given adjacency matrix. (pg. no.710, 10 E-book)
14. Graph Collaring and chromatic number
1. Define chromatic number.
2. What is the chromatic number of Kn? (pg. no.765, 10 E-book)
3. What are the chromatic numbers of the graph shown? (pg. no.764, 10 E-book)
4. Find the chromatic number of the given graph. (pg. no. 768, 10 E-book)
15. Isomorphism
1. Define Isomorphism.
2. Determine whether the graphs G and H displayed are isomorphic. (pg. no. 708 Ebook)
3. Are the simple graphs with the following adjacency matrices isomorphic? (pg. no.
713, 63 E-book)
4. Determine whether the given pair of directed graphs are isomorphic. (pg. no. 713, 69
E-book)
16. BFS and DSF
1. Define BFS and DFS with examples.
2. Use breadth-first search to find a spanning tree for the graph shown. (pg. no. 826 Ebook)
3. What is the output of depth-first search given the graph G shown? (pg. no. 831 Ebook)
4. Use breadth-first search to produce a spanning tree for the graph shown. (pg. no. 833,
13 E-book)
17. Kruskal's algorithm (MST)
1. Define Kruskal's algorithm.
2. Use Kruskal’s algorithm to find a minimum spanning tree for the weighted graph
shown. (pg. no. 839 E-book)
3. Use Kruskal’s algorithm to find a minimum spanning tree in the
weighted graph shown. (pg. no. 838 E-book)
4. Use Kruskal’s algorithm to find a minimum spanning tree in the
weighted graph shown. (pg. no. 839 E-book)
18. Prim’s algorithm
1. Define Prim’s algorithm.
2. Use Prim’s algorithm to find a minimum spanning tree for the weighted graph
shown. (pg. no. 839 E-book)
3. Use Prim’s algorithm to find a minimum spanning tree in the weighted
graph shown. (pg. no. 837 E-book)
4. Use Prim’s algorithm to find a minimum spanning tree in the weighted
graph shown. (pg. no. 839 E-book)