Time: 2 Hours Max.
Marks: 40
SECTION –A
(Answer all questions in one word or one
sentence each. Each question carries 1 mark)
1.
What is tautology?
2.
Define characteristic function.
3.
Convert the following English sentence to symbolic form using
Boolean variables and connectives : If I am not in good mood or I am not busy then
I will go for movie
4.
Give an example of a relation which is reflexive but is
neither symmetric nor transitive
(4*1=4)
SECTION –B
(Answer
any five questions in one
paragraph each. Each question carries 2 marks)
5.
If f(x)= [4-(x-7)3]1/5 is a real function then find f-1
6.
A function f is given as { (2,7) (3,4) (7,9)
(-1,6) (0,2) (5,3)} . Is this function one – one onto? Interchange the order of
the elements in the ordered pairs and form the new relation. Is this relation a
function? If it is a function is it one-one onto?
7.
Let A and B be two sets. If f:
A to B is one-one onto prove that f-1: B to A is also one-one and
onto.
8.
Define alpha cut set with an
example.
9.
Let C be the set of complex
numbers. Prove that map f: C to R given by f(z)=|z|, z in C is neither one-one
nor onto.
10.
Define a relation and a
function and give example to illustrate the difference between the two.
(5*2=10)
SECTION –C
(Answer any
four questions in not exceeding 120 words. Each question carries 3 marks)
11. In a committee 50 people speak French, 20 speak Spanish and
10 speak both Spanish and French. How many speak atleast one of these two
languages?
12. Let R be an equivalence relation. Prove
that R induces a partition.
13. If f : X to Y and g: Y to Z be two one-one
onto maps then (gof) : X to Z is also one-one and onto.
14. Let X={x: x in R and -
≤ x ≤
} and Y = {y: y in R and -1≤y≤}. Show that f:
X to Y defined by f(x) = sin x is one-one and onto. Also find the inverse of f.
15. Consider the set N x N the set of ordered pairs
of natural numbers. Let R be the relation in N x N defined by (a,b) R (c,d) if
and only if ad=bc. Prove that R is an
equivalence relation.
(4*3=12)
SECTION- D
(Answer any
two questions in not
exceeding four pages each. Each question carries 7 marks each)
16. State and prove inclusion exclusion
theorem.
17.
Test the validity of the argument
If I enter the poodle den, then I will carry my electric
poodle prod or my can of mace.
I am carrying my electric poodle prod but not my can of mace.
Therefore, I will enter the poodle den.
I am carrying my electric poodle prod but not my can of mace.
Therefore, I will enter the poodle den.
18. Write Warshall’s algorithm and use it to find
the transitive closure of
R=
{ (a,b), (b,d),(d,a),(d,c)}.
(2*7=14)
No comments:
Post a Comment