Set Theory and Algebra
Exam Duration: 45 Mins Total Questions : 25
What is the possible number of reflexive relations on a set of 5 elements?
- (a)
210
- (b)
215
- (c)
220
- (d)
225
Consider the set \(S=\left\{ 1,\omega ,{ \omega }^{ 2 } \right\} \) where \(\omega \) and \({ \omega }^{ 2 }\) are cube root of unity. If * denotes the multiplication operations, the structure {S*} forms
- (a)
a group
- (b)
a ring
- (c)
an integral domain
- (d)
a field
Which one of the following is true for any simple connected undirected graph with more than two vertices?
- (a)
Commutativity
- (b)
Associativity
- (c)
Existence of inverse for every element
- (d)
Existence of identity
Which one of the following is true for any simple connected undirected graph with more than two vertices?
- (a)
No two vertices have the same degree
- (b)
Atleast two vertices have the same degree
- (c)
Atleast three vertices have the same degree
- (d)
All vertices have the same degree
Consider the binary relation R={(x,y), (x,z), (z,x), (z,y)} on the set {x,y,z} which one of the following is true?
- (a)
R is symmetric but not anti-symmetric
- (b)
R is not symmetric but anti-symmetric
- (c)
R is both symmetric and anti-symmetric
- (d)
R is neither symmetric nor anti-symmetric
If P,Q,R are subset of the universal set U, then \(\left( P\cap Q\cap R \right) \cup \left( { P }^{ c }\cap Q\cap R \right) \cup { Q }^{ c }\cup { R }^{ c }\) is
- (a)
\({ Q }^{ c }\cup { R }^{ c }\)
- (b)
\(P\cup { Q }^{ c }\cup { R }^{ c }\)
- (c)
\({ P }^{ c }\cup { Q }^{ c }\cup { R }^{ c }\)
- (d)
\(\cup \)
Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are
- (a)
n and n
- (b)
n2 and n
- (c)
n and 0
- (d)
n and 1
Let X,Y,Z be sets of sizes X,Y and Z respectively. Let W=XxY and E be the set of all subsets of W. The number of functions from Z to E is
- (a)
Z
- (b)
Zx2xy
- (c)
Z2
- (d)
2xyz
The set {1,2,3,5,7,8,9} under multiplication modulo 10 is not a group. Given below are four plausible reasons. Which one of them is false?
- (a)
It is not closed
- (b)
2 does not have an inverse
- (c)
3 does not have an inverse
- (d)
8 does not have an inverse
Which one of the following option is CORRECT given three positive integers x,y and z and a predicate
- (a)
P(x) being true means that x is a prime number
- (b)
P(x) being true means that x is a number other than 1
- (c)
P(x) is always true irrespective of the value of x
- (d)
P(x) is being true means that x has exactly two factors other than 1 and x
Let Xn denotes the number of binary strings of length n that contains no consecutive 0's.
Which of the following recurrences does Xn satisfy?
- (a)
Xn =2Xn-1
- (b)
Xn =Xn/2 + 1
- (c)
Xn = Xn/2 + n
- (d)
Xn =Xn-1 + Xn-2
Let Xn denotes the number of binary strings of length n that contains no consecutive 0's.
The value of X5 is
- (a)
5
- (b)
7
- (c)
8
- (d)
None of these
Consider the set of the (column) vectors defined by X={x\(\epsilon \)R3|x1+x2+x3=0, where xT=[x1,x2,x2]T}Which of the following is true ?
- (a)
{1,-1,0]T, [1,0,-1]T is a basis for the surface X
- (b)
{[1,-1,0]T,[1,0,-1]T} is a linearly independent set
- (c)
X is not a sub place for R3
- (d)
None of these
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs (a,b) and (c, d) in the chosen set such that a \(\equiv \) c mod 3 and b\(\equiv \)d mod 5?
- (a)
4
- (b)
6
- (c)
16
- (d)
24
Let S={1,2,3,...,m}, m>3. Let X1,....,Xn be subsets of S each of size 3.Define a function f from S to the set of natural number as, f(i) is the number of sets Xj that contain the element i. i.e.,\(f(i)=\left| \{ j \right| i\epsilon { X }_{ i }\} |,\) then \(\sum _{ i=1 }^{ m }{ f(i) } \)is
- (a)
3m
- (b)
3n
- (c)
2m+1
- (d)
2n+1
Consider the set H of all 3\(\times\)3 matrices of the type
\(\left[ \begin{matrix} a & f & e \\ 0 & b & d \\ 0 & 0 & c \end{matrix} \right] \)
where, a,b,c,d,e, and f are real number and abc\(\neq \)0.Under the matrix multiplication operation the set S is
- (a)
a group
- (b)
a monoid but not group
- (c)
a semi-group but not a monoid
- (d)
neither a group nor a semi-group
Let G(x)=\(\frac { 1 }{ { (1-x) }^{ 2 } } =\sum _{ i=0 }^{ \infty }{ g(i){ x }^{ i } } ,\quad where\quad |x|<1,\quad what\quad is\quad g(i)?\)
- (a)
i
- (b)
i+1
- (c)
2i
- (d)
-2i
Let (S,\(\le \)) be a partial order with two minimal elements and a and b and a, c maximum element. Let P : S \(\Rightarrow\){True, False} be a predicate defined on S. Suppose that P(a) = True, P(b) = False and P(x) \(\Rightarrow\) P(y), \(\forall \)x, y\(\epsilon \)S satisfying x \(\le \)y, where \(\Rightarrow\) stands for logical implication .Which of the following statement cannot be true?
- (a)
P(x) = True for all x\(\epsilon \)S such that x\(\epsilon \)0 such that x\(\neq\) b
- (b)
P(x) = False for all x\(\epsilon \)S Such that x\(\neq\)a and x\(\neq\)c
- (c)
P(x) = False for all x\(\epsilon \)S Such that b\(\le \)x such tht x\(\neq\)c
- (d)
P(x) = False for all x\(\epsilon \)S Such that a\(\le \)x and b\(\le \)x
Consider the set {a,b,c} with binary operators + and x defined as follows
+ | a | b | c | x | a | b | c |
a | b | a | c | a | a | b | c |
b | a | b | c | b | b | c | a |
c | a | c | b | c | c | c | b |
e.g., a+c = c, c+a=a,c\(\times\)b = c and b\(\times\)c = a.
Given , the following set of equation :
(a\(\times\) x) + (a\(\times\)y) = c
(b\(\times\)x)+(c\(\times\)y) = c
The number of solutions (i.e., pairs (x,y) that satisfies the equation ) is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Let A, B and C be non - empty set and let x = (A-B)-C and Y =(A-C) - (B-C).Which one of the following is true?
- (a)
X = Y
- (b)
X\(\subset \) Y
- (c)
Y\(\subset \)X
- (d)
None of these
The set of {1,2,4,7,8,11,13,14} is a group under multiplication modulo 15. The inverse of 4 and 7 are respectively
- (a)
3 and 13
- (b)
2 and 11
- (c)
4 and 13
- (d)
8 and 14
Let E,F andG be finite sets . Let X = (E \(\cap \) F)-(F \(\cap \) G) and Y = (E-(E\(\cap \) G)-E-F).Which one of the following is true?
- (a)
XY
- (b)
XY
- (c)
X = Y
- (d)
X -Y\(\neq \)\(\phi \) and Y-X \(\neq \) \(\phi \)
The number of vertices of degree zero in G is
- (a)
1
- (b)
n
- (c)
n+1
- (d)
2n
The number of connected components in G is
- (a)
n
- (b)
n+2
- (c)
2n/2
- (d)
\(\frac{2^n}{n}\)
Let P,Q,R be three atomic propositional assertions.Let X denotes (P\(\vee \)Q) \(\rightarrow\) R and Y denotes (P\(\rightarrow\)R)\(\vee \)(Q\(\rightarrow\)R).Which one of the following is tautology?
- (a)
X \(\equiv \) Y
- (b)
X\(\rightarrow\)Y
- (c)
Y\(\rightarrow\)X
- (d)
\(\neg \)Y\(\rightarrow\)X