UNIT 1
SET , RELATION , FUNCTION AND THEOREM PROVING
TECHNIQUES
Unit-01/Lecture-01 
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 1-8}
PROBLEMS ON SET THEORY
Unit-01/Lecture-02
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 10-15}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Show that A Π
(B U C)= (AΠ B) U (AΠC) |
Dec2013 |
4 |
|
Q.2 |
If A, B, C are three sets Prove (A C) Π (B C) = (A Π B) - C |
June2009 |
10 |
|
Q.3 |
If U is a Universal set and A and
B are its subsets then Prove the following De Morgans laws a) (A
U B) = A Π
B b) (A Π
B) = A U B |
June 2013 |
7 |
|
Q.4 |
For any three Sets A, B and C
Prove that A X (B U C)= (A X B) U (A X C) |
June 2014 |
2 |
CARTESIAN PRODUCT OF TWO SETS
Unit-01/Lecture-03
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 31-34}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Prove that A X (B Π C) =( A X B) Π (A X C) |
Dec2013 |
4 |
|
Q.2 |
If A={1,4}, B= {4,5} and C=
{5,7}, determine i ) (AXB) U (AXC) ii) (AXB) Π
(AXC) |
Dec2015 |
2 |
INCLUSION, EXCLUSION PRINCIPLE
Unit-01/Lecture-04
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 21-29}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Among 100 students 32 study
mathematics, 20 study Physics,45 study Biology, 15 study mathematics and
Biology, 7 study mathematics and physics, 10 study physics and Biology,and 30
donot study any of these subjects- (i)
Find the number of students studying all these subjects. (ii)
Find the number of students studying exactly one of these
subjects. |
Dec2012, Dec 2013, June
2014 |
7 |
|
Q.2 |
Out of a total of 130 students 60
are wearing hats to class, 51 are wearing scarves and 30 are wearing
both hats and scarves. Of the 54
students who wearing sweaters, 26 are wearing hats, 21 are wearing scarves
and 12 are wearing both hats and scarves. Everyone wearing neither a hat nor
a scarf is wearing gloves. (i)
How many students are wearing gloves. (ii)
How many students not wearing a sweater are wearing hats
but not scarves? |
June 2007 |
7 |
MATHEMATICAL INDUCTION
Unit-01/Lecture-05
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 77-86}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Prove 12+ 22+
32+42+..........n2 = (1/6)n (n+1)(2n+1)
forall n N |
June 2005 |
7 |
|
Q.2 |
72n+23n-3.
3n-1 is divisible by 25 forall n N. |
June 2009, June 2014 |
10, 7 |
|
Q.3 |
11n+2+122n+1
is divisible by 133, n N |
June 2007 |
7 |
|
Q.4 |
Use MI to prove that 2.7n + 3.5n-5 is
divisible by 24 for all n >0 |
Dec 2011, June 2011 |
7 |
|
Q.5 |
Prove by induction that 12+ 32+ 52
+....+ (2n-1)2 = n(2n+1)(2n-1)/3 = 1/3n (4n2-1) |
June 2013 |
7 |
|
Q.6 |
Prove 13+ 33+53+..........(2n-1)3
= (1/3)n (2n+1)(2n-1) for all n N |
Dec2015 |
7 |
PIGEON HOLE PRINCIPLE
Unit-01/Lecture-06
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 73-76}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Explain Pigeonhole principle? |
June 2011 |
2 |
|
Q.2 |
Prove that if n pigeons are
assigned to m pigeonholes (n>m), then some pigeonhole must contain atleast
(n-1)/m +1 pigeons |
June 2009 |
10 |
|
Q.3 |
If a company has 13 employees
then show that two of them were born in the same month |
Dec 2012 |
7 |
|
Q.4 |
Prove that among 100,000 people
there are two who are bon at same time? |
June 2007 |
7 |
|
Q.5 |
Write Short note on Relation. |
June 2011 |
2 |
|
Q.6 |
Briefly explain the application
of Pigeon hole principle using an example. |
Dec2015 |
3 |
REPRESENTATION OF RELATION ON FINITE SETS, EQUIVALENCE
RELATION
Unit-01/Lecture-07
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 36-41}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Define Equivalence Relation. |
Dec 2012 |
2 |
|
Q.2 |
Let f: R →R be defined as
Let g: R→R be defined as
Then find the
composition gof. |
Dec2015 |
7 |
PROBLEMS ON RELATIONS
Unit-01/Lecture-08
PARTIAL ORDER RELATIONS, FUNCTIONS OR MAPPINGS
Unit-01/Lecture-09
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL,41-44}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Show that the relation R= {
(a,b): a-b =even integer and a,b I
} in the set I of integers is an equivalence relation |
Dec 2011, June 2011, June
2014 |
7, 2 |
|
Q.2 |
Let R be a binary relation on the
set of all positive integers such that R= { (a,b): a-b =odd integer and a,b I } Is R reflexive? Symmetric? Anti
Symmetric? Equivalence Relation? |
Dec 2008 |
10 |
|
Q.3 |
Show that the relation R= {
(a,b): a-b =a,b I and
a-b is divisible by 3} is an
equivalence relation |
Dec 2013 |
7 |
|
Q.4 |
If R and R are equivalence
relations in a set A show that R Π R is an equivalence relation in A |
June 2007 |
7 |
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 55-67}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Let R and R be any two partial
order relations . Then show that R Π R is also a partial order
relation |
June 2007 |
7 |
|
Q.2 |
Explain partial ordering relation |
June 2011 |
2 |
|
Q.3 |
Explain Function |
June 2011 |
2 |
|
Q.4 |
Short notes on- (i)
Composite of mappings (II) Inverse Mappings |
June 2011 |
7 |
PROBLEMS ON FUNCTIONS
Unit-01/Lecture-10
REFERENCES {DISCRETE STRUCTURES BY DR.D.C
AGARWAL, 55-67}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Show that the mapping F: N-> N
defined by f(x)= x2 is one one into where N is a set of natural
numbers. |
Dec 2011 |
7 |
|
Q.2 |
Write short note on Recursively
Defined functions |
June 2011 |
2 |
|
Q.3 |
Let X and Y be two sets. If a
mapping f: X-> Y is one one onto prove that f-1 is also one one
onto |
June 2014 |
3 |