Sl No	Questions
1.	What do you mean by a set?  Give three examples to substantiate your answer. A set is a collection of objects called elements, which share some common properties. Examples   (i) The set of Natural numbers  (ii) The set of Whole numbers (iii) The set of integers (iv) The set of real numbers etc. A set is a collection of objects called elements, which share some common properties. Any object  belonging to a set is called an element of that set. The elements in a set are like: People, building, car, letters etc.
2.	Define the following with an illustrations to each
	(a) Universal set: A set is called a universal set if it includes every set under discussion.  A universal set is denoted by U
	(b) Null set: A set which does not contain any element is called an empty set or a null set denoted by j
	(c) Equavalent sets:
3.	Define the union and intersection of two sets. Give an example
	Let A and B be any two sets.  The union between these two sets, denoted by A  B is the set of all elements which are members of the set A or the set B or both.  Symbolically, it is written as A È B = {x/xεA v xεB v both} A È A = A
4.	Define
	(a)    Relative complement of the set A with respect to the set B Let A and B be any two sets.  The relative complement of B in A (or of B with respect to A) written A – B is the set consisting of all elements of A which are not elements of B, i.e.,          A – B = {x/x ε A L x Ï B }. The relative complement of B in A is also called the difference of A and B
	(b)   Complement of a set A (c)    Let U be the universal set.  For any set A,  the relative complement of A with respect to U, i.e., U – A is called the (absolute) complement of A and is denoted by Ac, i.e., (d)            Ac= U – A = {x/x ε U L x Ï A }.
5.	If Â={-¥, ¥}, A={x/x is a negative integers}, B={x/x is a nonpositive integer}
	Find (i )AÈÂ (ii) AÇÂ and (iii) AÇB, (iv) Ac (v) A-B
6.	Define the symmetric difference between the two sets Let A and B be any two sets.  The symmetric difference of A and B is the set consisting of all the elements that belong to A or B but not to both A and B.  It is dented by A Δ B i.e., A Δ B = {x/(xεA L xÏB) or (xεB L xÏA}.           = (A – B) U (B – A)
7.	If A={1, 2, 3, 4, 5, 8} and B={1, 2, 3, 5, 6, 7} find the symmetric difference the set A and the set B.
8.	If A, B, and C are any three sets prove that (i) (AÈB)ÈC = AÈ(BÈC) and (ii) (AÇB)ÇC = AÇ(BÇC)
9.	State De Morgan’s law for a swequence of n sets A1, A2, A3, …, An
10.	Mention the properties of Union, Intersection, Complement, and Symmetric difference between the sets. Let A, B, and C are sets and U be the universal set. Then   (i) Ac = U - A  (ii) A – B = A ∩ Bc  (iii) A – A = j (iv) A – j = A  (v) A – B = B – A which implies A = B (vi) A – B = A which implies A ∩ B= j (vii) A – B = j which implies AÍB Symmetric difference Let A and B any sets, then   (i) AΔA = j  (ii) A Δj = A  (iii) A Δ B = B Δ A (iv) A Δ B = (AÈB) –(A ∩ B)
11.	Represent the following using the Venn diagram
	(i) AÈB, (ii) AÇB, (iii) AÇBc, (iv) (AÈB)c, (v) (AÇB)c
12.	Define the Cartesian product of n-tuples. the Cartesian product of n sets A1, A2, …, An is defined as A1x A2x …x An = {(a1, a2, …, an)/ai ε Ai, for i = 1, 2, …, n}. The expression {(a1, a2, …, an) is called an ordered n-tuple.
13.	Let A, B, And C are finite sets.  Prove that
	(i) A X (BÈC) = (A X B) È (A X C)
	(ii) A X (BÇC) = (A X B) Ç (A X C)
14.	If A={1, 2, 4}, B={2, 5, 7}, and C={1, 3, 5} are any three sets verify
	(i) A X (BÈC) = (A X B) È (A X C)
	(ii) A X (BÇC) = (A X B) Ç (A X C)
15.	Show by means of an example
	(i) A X B ¹ B X A
	(ii) (A X B) X C = A X( B X C)
16.	Define (i) binary relation, (ii) Domain of a relation, and (iii) Range of a relation
17.	Mention the properties of relations.
18.	If R is the set of all real numbers and ° is the binary operation on R defined by a ° b=2a +3b, (a, b) Î R, show that ° is neither associative nor commutative.
19.	Examine whether the operatins ‘Å’ and ‘Ä’ defined by a Å b = a+2ab,  aÄ b =  an, for positive integers, obey the commutative, associative and distributive laws on not.
20.	Let Q be the set of rationals and an operation ° is defined over Q - {1} as a ° b = a + b – a.b, ‘+’ and ‘.’ being ordinary addition and multiplication.  Verify which of the algebraic laws satisfy the operatin ° over the set Q – {1}.
21.	For the relations R and S define (i) RÈS, (ii) RÇS, (iii) R-S, and (v) Rc
22.	For the relations R and S define the composite of (i) R and S and (ii) S and R.  Are they commutative?
	Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 2), (3, 4), (4, 1)}.  Is R (i) Reflexive, (ii) Symmetric,
	(iii) Antisymmetric, (iv) Transitive?
23.	If A={1, 2, 3}, B={1, 2, 3, 4} and C={0, 1 , 2}, the relations R and S are defined as
	R={(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)}
	S={(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}
	Find SoR. Can we find RoR? If so find RoR.
24.	Let X={1, 2, 3, 4, 5, 6, 7} and R={(x, y)/x-y is divisible by 3} in X.  Show that R is an equivalence relation.
25.	Let Z be the set of integers and let R be the relation called “congruence modulo 3” defined by
	R={(x, y)/x e Z Ù y e Z Ù (x-y) is divisible by 3}.  Is R an equivalence relation?
26.	Let A={v1, v2, v3, v4, v5}. Let R be a relation defined by R={(v1, v2), (v1, v3), (v3, v5), (v4, v1), (v4, v4), (v5, v2), (v5, v3), (v5, v1)}.  Write the adjacency matrix of R and draw a digraph for R.
27.	Define the following terms
	(i) Into function
	(ii) One-to-one function
	(iii) Bijective function
	(iv) Many-one mapping
28.	Which of the following relations on the set A = {1, 2, 3} is (i) reflexive, (ii) sy;mmetric, (iii) transitive, (iv) antisymmetric?
	R1 = {(1,1), (1, 2), (1, 3), (3, 3)}
	R2 = {(1,1), (1, 2), (2, 1), (2, 2), (3, 3)}
	R3 = {(1,1), (1, 2),  (2, 2), (3, 3)}
29.	Let A = {1, 2, 3, 4} and R = {(1,1), (1, 2), (2, 1), (2, 2), (3, 4), (3, 3) (4, 3), (4, 4) } be a relation on A.  Verify that R is an equivalence relation.
30.	On the set ;of all integers Z, consider the relation R = {(a, b)ç a º r (mod 2) and b º r (mod 2) } where r is a fixed integer.  Prove that R is an equivalence relation.
31.	Consider the sets A = {a, b, c} and B = {1, 2, 3} and relations R = {(a, 1), (b, 1), (c, 2), (c, 3)} and S = {(a, 1), (a, 2), (b, 1), (b, 2)} from A to B.  Determine
	(i) Rc, (ii) Sc, (iii) RÈS, (iv) RÇS, (v) R-1, (vi) S-1
32.	Let R be a relation from A to B.  Prove that
	(i) dom (R-1) = ran (R) (ii) ran(R-1) = dom (R)
33.	Let R and S be relations from set A to a set B, then the follwoing is true:
	(i) If R Í S, the R-1 Í S-1
	(ii) If R Í S, the Sc Í Rc
	(iii) (R Ç S)-1 = R-1 Ç S-1
	(iv) (R È S)-1 =  R-1 ÈS-1
	(v) (R ÇS )c = Rc ÈSc
	(vi) (R ÈS)c = Rc ÇSc
34.	Let R be a relation on a set A.  Then the following are true:
	(i) If R is reflexive, so is R-1
	(ii) R is reflexive if and only if Rc is reflexive
	(iii) R is sym;metric if and only if R = R-1
	(iv) If R is symmetric, so are R-1 and Rc
	(v) R is antisymmetric if and only if RÇR-1 = j
35.	Let R and S be relations from set A to a set B, then the follwoing is true:
	(i) If R and S are reflexive, so are R È S and R Ç S
	(ii) If R and S are symmetric, so are R È S and R Ç S
	(iii) If R and S are transitive, so is R Ç S
37.	Let R and S be relations from set A to a set B, then the follwoing is true:
	(i) If R is an equivalence relation, so is R-1
	(ii) If R and S are equivalence relations, so is R Ç S
38.	Let A, B, and C are three sets, R be a relation from A to B and S be a relation from B to C.  Prove that (SoR)-1 = R-1 o S-1
39.	Show that the composition of relations is associative.
40.	Let R = {(1, 2), (2, 2), (3, 4)} and S = {(1, 3), (2, 5), (3, 1), (4, 2)} be relations on the set A = {1, 2, 3, 4, 5}. Find (i) SoR, (ii) RoS, (iii) RoSoR, (iv) SoRoS
41.	Let X = { xçx is real and x ³ -1} and Y = { xçx is real and x ³ 0}. Consider the function f: A®B defined by f(x) = Ö(a+1), for all xÎX.  Show that f is invertible and determine f-1
42.	Let X = Y= Z = R, be the set of all real numbers and f:X®Y and g: Y®Z be defined by
	f(x) = 2x + 1 and g(y) = 1/3y, "x Î X and " y Î Y.  Compute gf and show that (gof)  is invertible.  What is (gof)-1?
43.	Let X = Y = R, be the set of all real numbers and the functions f:X®Y and g:Y®X defined by
	f(x) = 2x3 – 1 " xÎ X; g(y) = {½(y  + 1)}1/3 " y Î Y.  Show that each of f and g is a one-to-one correspondence and that each is the inverse of the other.
44.	Define an Abelian group.  Is the set of integers Z an Abelian group?
45.	Prove that the identity element and the inverse element in a group áG, *ñ are both unique.
46.	Let G be the set of real nu;mbers and not equal to -1 and * defined by a * b = a + b +ab.  Prove that áG, *ñ is an Abelian group.
47.	Let G be a group with identity element e.  Show that if x2 = x for some x in G, then x = e
48.	In a group g having more than one element, if x2 = x for every xG, prove that G is an Abelian group
49.	Prove that a group G in which every element is its own inverse in Abelian group.
50.	Prove that a group G is Ableian if and only if (ab)2 = a2b2 for all a, b Î G
51.	Consider the additive group áZ, +ñ of all integers and the group {-1. 1}under multiplication. Define f: Z® H by -1, if n is odd                           f(n)= Prove that f is homomorphism.
52.	Prove that the set S={0, 1, 2, 3, 4, 5} forms a ring under addition and multiplication modulo 6. Is this forms a field?
53.	Define the following terms (i) Random experiments, (ii) Sample space, (iii) Mutually exclusive outcomes, (iv) Equally likely outcomes An experiment in which results are unpredictable The set of all possible outcomes of a random experiment is called sample space denoted by W. Occurrence of one event precludes or prevents the occurrence of other event in a random experiment Every outcome of a random experiment which occur will have an equal chance
54.	Define probability in the classical approach.  What are its limitations? In a random experiment, out of ‘n’ exhaustive, mutually exclusive, equally likely, independent events if ‘m’ of them are favorable to the occurrence of an event, say, ‘A’, then the probability of an event ‘A’, denoted by P(A) is m/n 1.This definition emphasizes that the events must be equally likely. Thus, fails when various  outcomes of a trail are not equally likely.  2. This definition is useful only when we deal with card games, dice games, coin tossing and the like.  It fails in situations when we try to apply it to less orderly decision problem we encounter in management. 3. It does not consider those situations that are unlikely but that could conceivably happen.  Like the occurrence of a coin landing on its edge or our room burning watching TV etc., which are extremely unlikely but not impossible. 4. In case the exhaustive number of cases in a trial is infinite, the definition fails to give the required probability. 5. In some situation there may be a difference of opinion with respect to the ways of forming the possible and favourable outcomes.
54.	Give the axiomatic definition of probability. Axiom 1: P(A) is a real number, i.e., P(A) ³ 0 for every A e S or 0£ P(A) £1. Axiom 2:  P(W) = 1. We call P(A) the probability of an event.  The axiom 3 is called countable additivity. Thus, if W is a sample space consisting of n events and if each event is equally likely to happen, then its probability is 1/n.  Now if A is an event of W containing m elements of  W, i.e., n(A) = m, then P(A) = 1/n + 1/n + 1/n +…+ up to m              terms         = m/n
55.	State and prove addition rule of probability.
56.	Prove that (a) (b)
57.	Define the conditional probability of two events.
58.	A box contain 6 red, 4 white and 5 black balls.  If 4 balls are drawn at random what is the probability that among the balls drawn there is at least one ball of each colour?
59.	Two dice, say one green and the other red, are thrown.  Let A be the event that the sum of the points on the faces shown is odd and B be the event that at most one ace (number 1). Find the     probabilities of the events: (i) P(A), (ii) P(B), (iii) P(AÈB), (iv) P(AÇB), (v) P(AÇBc), (vi) P(AcÇBc),  and(vii) P (AÈB)c
60.	In a group of 160 Engineering students, 92 are enrolled for advanced course on CSD, 63 students on VLSI and 40 are enrolled in both.  How many of these students are not enrolled in either course?
61.	For any two events A and B prove that P(AcÇ B) = P(B) - P(A Ç B)
62.	For any two events A and B prove that P(AÇBc)ÈP(AcÇB)=P(A)+P(B)-2P(AÇB)
63.	In a group of 160 Engineering students, 92 are enrolled for advanced course on CSD,  63 students on VLSI and 40 are enrolled in both.  How many of these students are not enrolled in either course?
64.	In a city it was observed that 80% of    the families owns a two wheeler and 43% owns a car. Those who owns both are 38%.  If a family is selected at random what is the probability that they own either a two wheeler or a car.
65.	Prove that the probability of an impossible event is zero, i.e., P(j) = 0. The probability of an impossible event is zero, i.e., P(j) = 0 It follows that                W È j  = W            P(W È j) = P(W)            P(W) + P(j) = P(W) from axiom 3            P(j) = 1-1 = 0 from axiom 1           \P(j) = 0

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