DISCRETE MATHEMATICS(PART-01)

OBJECTIVE GENERAL KNOWLEDGE -  DISCRETE MATHEMATICS

1. Let R be the equivalence relation on the set of real numbers such that aRb if and only if a-b is an integer. Then equivalence class of [2]R is :
1. 2n, n belongs to R
2. Z
3. R , the set of real numbers
4. (2n+1)/2, n belongs to integers
2. Let R be the equivalence relation on the set of real numbers such that aRb if and only if a-b is an integer. Then equivalence class of [1/2]R is :
1. 2n, n belongs to R
2. Z
3. R , the set of real numbers
4. (2n+1)/2, n belongs to integers
3. Consider the function f(x) = x3 + 1, then f is
1. Onto 
2. One to one
3. Both one to one and onto
4. Neither one to one nor onto
4. Let R be the relation on the set of  people consisting of pairs (a, b), where a is a parent of b. Let S be the relation on the set of people consisting of pairs (a, b), where a and b are siblings (brothers or sisters). What is and R ◦ S?
1. R ◦ S = {(a,b)| b is an uncle or aunt of a}
2. R ◦ S = {(a,b)| a is an uncle or aunt of b} 
3. R◦ S = {(a,b)| b is parent of a}
4. R ◦ S = {(a,b)| a is parent of b}
5. Let R =ϕ be a relation on a nonempty set S. Then R is (a) reflexive(b) symmetric(c) transitive(d) anti - symmetric
1. Both b and c
2. Both a and c
3. d
4. Both a and d
6. Let R be the relation on the set of ordered pairs of all real numbers such that ((a, b), (c, d))∈ R if and only if ad = bc.  What is the equivalence class of (2, 3)?
1. {(a, 3a/2) , a belongs to Real numbers}
2. {(1, 3/2)}
3. {(a, 2a/3) , a belongs to Real numbers}
4. {(2a, 3a/2) , a belongs to Real numbers}
7. Suppose that the relation R is irreflexive. Is R2 necessarily irreflexive?
1. True
2. False
8. The set of all congruence class modulo 3 is given by
1. { [1]3 , [2]3 , [4]3 }
2. { [0]3 , [1]3 , [2]3 }
3. { [0]3 , [1]3 , [3]3 }
4. { [0]3 , [1]3 , [4]3 }
9. The 12 th term in the sequence corresponding to the generating function (1-x)-2 is
1. 144
2. 11
3. 12
4. 13
10. Which of the following represent the sequence 1, 2, 5, 11, 26,.....
1. t( n ) = 2t(n-1)+2, t(0) =1, t(1)=2
2. t( n ) = t(n-1)+t(n-2), t(0) =1, t(1)=2
3. t( n ) = 2t(n-1)+1, t(0) =1, t(1)=2
4. t( n ) = t(n-1)+3t(n-2), t(0) =1, t(1)=2
11. If x=uv is a bridge for a connected graph G which is not a complete graph of 2 vertices, then neither u nor v is a cut point of G.
1. True 
2. False
12. If G contains no M-augmenting path, then M is
1. perfect matching and maximum matching
2. Perfect matching
3. Maximum matching
4. Not a matching
13. The number of vertices and edges in Km,n are
1. m+n,mn
2. m+n,m+n
3. mn,m+n
4. m,n
14. The number of edges present in a complete graph having n vertices is
1. Information given is insufficient
2. n
3. n(n-1)/2
4. n(n+1)/2
15.  The chromatic number of Wn  ,if n is odd is 
1. 2
2. 5
3. 3
4. 4
16. The poset (Z+,/)
1. Totally ordered
2. Set
3. Incomparable
4. Totally ordered and incomparable
17. Every interval of a lattice is a sublattice :
1. True
2. False
3. Not always true
18. For a poset (p,≤) its dual (p,≥) is also a poset . The least member of p relative to the ordering ≤  is the ---------- member in p relative to the ordering ≥
1. Greatest
2. Least
3. Neither greatest nor least
19. In a connected graph every line can be a bridge.
1. True
2. False
20. A complete graph with six vertices has a perfect matching
1. True 
2. False
21. The union of any two distinct u-v paths contains a cycle.
1. True
2. False
22. Any connected bi-partite graph is uniquely 2- colourable
1. True 
2. False
23. Every lattice with 0 and 1 is complemented
1. True
2. False
24. If L is a distributed lattice with 0 and 1, then each element x∈Lx∈L has
1. No complement
2. Atleast one complement
3. Atmost one complement
25. Mention the lattice which is called a Boolean algebra or a Boolean lattice.
1. Distributive
2. Modular
3. Complemented
4. Complemented distributive 
26. If D(n) denotes the lattice of all positive divisors of a positive integer n . Determine how many sublattices of D(45) exists with exactly five elements.
1. 1
2. 3
3. 2
4. 4
27. The poset P({a,b,c},⊆)P({a,b,c},⊆)
1. Totally ordered set
2. Not totally ordered set
28. For what values of n does the complete graph Kn have perfect matching?
1. Odd
2. Even
3. Prime
4. Odd or even
29. If n is a positive integer and p2/n , when n is a prime number, then D( n ) is a boolean algebra.
1. True
2. False
30. Every element x in a Boolean algebra has
1. More than one complement
2. Unique complement
3. No complement