1.

2.

3.

4.

5.

6.

7. Determine the sequence of corresponding to each of the following generating function:

8. Determine the sequence of corresponding to each of the following generating function:

9. Determine the sequence of corresponding to each of the following generating function:

(1+x)/(1-x)(1-3x).

10.

11.

12.

13.

14.

15.

16. Let 𝑓 ∶ 𝑅 → 𝑅 defined as 𝑓(𝑥) = 2𝑥 + 3. Investigate 𝑓 is bijective or not. If 𝑓 is bijective

then find inverse of 𝑓.
17. Let 𝑓 ∶ 𝑅 → 𝑅 defined as 𝑓(𝑥) = 5𝑥 + 4. Investigate 𝑓 is bijective or not. If 𝑓 is bijective
then find inverse of 𝑓.
18. Let 𝑓 ∶ 𝑅 → 𝑅 defined as 𝑓(𝑥) = 3𝑥 + 7. Investigate 𝑓 is bijective or not. If 𝑓 is bijective
then find inverse of 𝑓.

19.

20.

21. Determine the adjacency matrix of the following graph

A.

B.

22. Find 𝑢 and 𝑣 such 𝑔𝑐𝑑(252, 70) = 252𝑢 + 70𝑣.

23. Find integers 𝑢 and 𝑣 such 𝑔𝑐𝑑(252, 334) = 252𝑢 + 334𝑣.
24. If 𝑔𝑐𝑑(𝑎, 𝑏) = 1, prove that 𝑔𝑐𝑑 (𝑎 + 𝑏, 𝑎𝑏) = 1.
25. If 𝑎|𝑐 and 𝑏|𝑐 with 𝑔𝑐𝑑 (𝑎, 𝑏) = 1, prove that 𝑎𝑏|𝑐.
26. If 𝑔𝑐𝑑(𝑎, 𝑏) = 1, prove that 𝑔𝑐𝑑 (𝑎2, 𝑏) = 1.
27. Prove that 𝑝 ↔ 𝑞 ≡ (𝑝 → 𝑞) ∧ (𝑞 → 𝑝).
28. Show that the proposition (p ∧ q) ∧~ (p ∨ q) is contradiction.
29. Show that the proposition formula (~q ∧ p) ∨~ (p ∧ q) is tautology.
30. Find the order of each element of (Z6, +).
31. Find the order of each element of (Z5, +).
32. Show that if every element of a group (G, o) be its own inverse, then it is a
commutative/abelian group. Is the converse true?
33. Show that the following two graphs are isomorphic.

A.

B.
 34. For any three sets A, B, C, show that 𝐴 ∪ (𝐵 ∪ 𝐶) = (𝐴 ∪ 𝐵) ∪ 𝐶.
35. For any three sets A, B, C, show 𝑡ℎ𝑎𝑡 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶).

Multiple Choice Questions

1. Which of the following is a tautology?

a) p∧¬p

b) p∨¬p

c) p→¬p

d) 𝑝 ∧ 𝑞

2. Which of the following is the contrapositive of 𝑝 → 𝑞?

a) ¬p→¬q

b) ¬q→¬p

c) p→¬q

d) 𝑞 → 𝑝

3. According to the Division Algorithm, for any integers a and b where b>0, there exist unique
integers q and r such that:

a) 𝑎 = 𝑏𝑞 + 𝑟 𝑎𝑛𝑑 0 ≤ 𝑟 < 𝑏.

b) 𝑎 = 𝑏𝑞 − 𝑟 𝑎𝑛𝑑 0 ≤ 𝑟 < 𝑏.

c) 𝑎 = 𝑏𝑞 − 𝑟 𝑎𝑛𝑑 0 ≤ 𝑟 ≤ 𝑏.

d) 𝑎 = 𝑏𝑞 + 𝑟 𝑎𝑛𝑑 0 ≤ 𝑟 ≤ 𝑏.

4. Given a=17 and b=5, what are the values of q and r according to the Division Algorithm?

a) 𝑞 = 3, 𝑟 = 2

b) 𝑞 = 4, 𝑟 = 3

c) 𝑞 = 2, 𝑟 = 3

5. Which of the following is a valid identity for sets?

a) 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)

b) 𝐴 ∪ (𝐵 ∩ 𝐶) = 𝐴 ∩ (𝐵 ∪ 𝐶)

c) 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)

d) 𝐴 ∩ (𝐵 ∪ 𝐶) = 𝐴 ∪ (𝐵 ∪ 𝐶)

6. If 𝐴 = {1,2} and 𝐵 = {𝑥, 𝑦}, the Cartesian product A×B is:

a) {(1, 𝑥), (2, 𝑦)}

b) {(1, 𝑥), (1, 𝑦), (2, 𝑥), (2, 𝑦)}

c) {(1, 𝑥)}

d) {(𝑥, 1), (𝑦, 2)}

7. A relation R on a set A is called reflexive if:

a) (𝑎, 𝑏) ∈ 𝑅 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 (𝑏, 𝑎) ∈ 𝑅

b) (𝑎, 𝑎) ∈ 𝑅 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑎 ∈ 𝐴

c) (𝑎, 𝑏) ∈ 𝑅 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑎, 𝑏 ∈ 𝐴

d) (𝑎, 𝑏) ∈ 𝑅 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 (𝑏, 𝑐) ∈ 𝑅 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐 ∈ 𝐴

8. A function 𝑓: 𝐴 → 𝐵 is bijective if:

a) It is both one-to-one and onto

b) It is one-to-one but not onto

c) It is onto but not one-to-one

d) It is neither one-to-one nor onto

9. If 𝑓: 𝐴 → 𝐵 and 𝑔: 𝐵 → 𝐶, the composite function 𝑔 ∘ 𝑓 is:

a) A function from A to C.

b) A function from B to A

c) A function from C to A

d) A function from C to B

9. If 15 pigeons are placed in 12 pigeonholes, then the number of pigeonholes that contain at
least 2 pigeons is:
a) 3

b) 4

c) 5

d) 6

10. A set S with a binary operation * is a monoid if:

a) Every element has an inverse under the operation.

b) The operation is associative and there exists an identity element.

c) The operation is commutative and associative.

d) The operation is associative but there is no identity element.

11. Which of the following is a necessary condition for a set S with a binary operation ∗ to be a
group?

a) Closure and associativity only

b) Closure, associativity, and the existence of an identity element

c) Closure, associativity, and the existence of inverses for all elements

d) Closure and the existence of an identity element

12. Which of the following is an identity in Boolean algebra?

a) A+A=A

b) A×A=1

c) A+1=A

d) A×0=1

13. Which of the following is not a property of a simple graph?

a) No loops

b) No multiple edges

c) The degree of every vertex is even

d) None of these

14. In a graph, the degree of a vertex is defined as:

a) The number of vertices adjacent to it.

b) The number of edges incident to it.

c) The number of edges in the entire graph.

d) The total number of vertices in the graph

15. A graph is connected if:

a) There is a path between any two vertices.

b) Every vertex has a degree greater than zero.

c) It contains no cycles.

d) It contains at least one edge.

16. The symmetric group S3 is

a) cyclic but not abelian

b) cyclic and abelian

c) non-cyclic and non-abelian

d) none

17. The proposition p∧(q∧∼q) is a

a) contradiction

b) tautology

c) both (a) and (b)

d) none

18. (~(~(~p)))≡

a) p

b) ~(~p)

c) Tautology

d) None of these

19. If p:’Anil is rich’ and q:’Kanchan is poor’ then the symbolic form of the statement ‘Either Anil
 or Kanchan is rich’ is

a) 𝑝 ∨ 𝑞

b) 𝑝 ∨∼ 𝑞

c) ∼ 𝑝 ∨ 𝑞

d) 𝑝 ∧∼ 𝑞

20. If a is prime to b and a is prime to c then a is prime to

a) b2 and c2

b) 𝑏𝑐

c) 𝑏 + 𝑐

d) b2 + c2

21. In the set of integer the relation ρ is defined by 𝑎𝜌𝑏 hold if b is a multiple if a. Then ρ is

a) Reflexive

b) anti symmetric

c) transitive

d) equivalence

22. In Z3, [2]×[2]=?

a) 6

b) 4

c) 0

d) None of these

23. The remainder when 310 is divided by 7 is

a) 0

b) 1

c) 4

d) 7

24. The number of unit element of the ring ℤ is

a) 2

b) 3

c) 1

d) infinite

25. Which of the following is a subgroup of (ℤ, +)

a) set of all integers multiple of 3

b) set of all odd integers

c) set of all prime integers

d) { 1, −1 }