ASSIGNMENT-1


Que.1. Determine partial orderings on set {1,2,3}
(a){(0,0),(1,1),(2,2),(3,3)}
(b){(0,0),(1,1),(2,0),(2,2),(2,3),(3,2),(3,3)}


Que.2. Determine Posets:-
a)(z,=)
b)(z,>=)
c)(z,<=)


Que.3. Is (s,r) a poset if s is set of ale people in world & (a,b)ER, where a&b are people if :
(i)a is taller than b
(ii)a=b or a is an ancestor of b?


Que.4. Find equals of poset:-
(a)({0,1,2}<=)
(b)(z+,/)


Que.5.find incomparable elements of these posets :-
(p({0,1,2})<=)


Que.6. Consider a Poset ({3,5,9,15,24,45},/) , Find
1.maximal elements
2.minimal elements
3.greatest elements
4.least elements
5.Upper bound of {3,5}
6.Lowr bound of {15,45}


Que.7. What is coveling relation of partial ordering {(a,b)/ac=b} on power set of s,where s={1,2,3}?


ASSIGNMENT-2




Que.1. Draw hasse diagram of when d60 is set of all division of 60?


Que.2. Define lattice?


Que.3. Define isomorphic lattice?


Que.4. Consider Poset (a,1) where a={1,2,3,4,6,9,12,18,36} ordered by divisibility &poset (b,1) where b={2,4,6,8}
(i)draw hasse diagram for both poset.
(ii)Find maximal & minimal element in both.
(iii)Find supremum & infimum of every pair of elements in both posets.
(iv)Are these posets lattices?


Que.5. Show that product of two lattices is a lattices.


Que.6. Let S={1,2,3,...12} be poset under divisibility relations. Draw hasse diagram & find first & last element . Also find upper bound, lower bound for subset {5,7,8}.




ASSIGNMENT-3




Que.1. Draw simplified network of f(x,y,z)=x.y.z+x.y'.z+x'.y'z


Que.2. Express the following:- xy'+xz+xy in CNF as well as DNF;


Que.3. Simplify using K Map.Also draw circuit of simplify expression f(a,b,c,d)= Em (0,1,4,5,6,8,9,12,13,14)


Que. 4. Give CNF of expression (y+z') of three variables x,y,z.


Que.5. Define DNF find DNF for (a.b')+(b.c')+(c.a')


Que.6. For every element a&b in Boolean alzebra show that


(i) (a.b)'= a'+b'
(ii) (a+b)'=a'b'
.


ASSIGNMENT-4




Que.1. Show that (p^(-pvq))v(q^~(p^q))=q


Que.2. Write converse, inverse & contrapositive of following statements:
(i)if teacher is absent, then some students do not complete their homework.
(ii)All students complete their homework if they don not have a test


|Que.3. Show that
(p->(q^r))->(-r->-p) is a tantology.


Que.4. Test the validity of following arguments "If i enjoy studying ,then I will study . I will do my homework or I will not study. I will not do my homework. Therefore I do not enjoy studying."


Que.5. Show that (p->q)^(r->q)=(pvr)->q



ASSIGNMENT-5


Que.1. Find explicit formula for fibonacci numbers
(Since fn= fn-1 + fn-2 and f0=0,f1=1)


Que.2. Solve the following uncurrence relation
(i)an+2-6an+2+8an=3n2+2-d.3n
(ii)an+3-3an+2-an=24n+48


Que.3. Solve using generating function:-
(i)an-2an-1-3an-2=0
(ii)ar=2an-1+3


Que.4. Define pigeon hole principle.Find minimum no. of boys in the same minute out of 3000 boys on a day.


Que.5. In MCA class of 40 students ,5 are weak. Determine how many ways we can makes a group of students;


(i)Five good students
(ii)Five students in which exactly 3 are weak.


Que.6. Find complete column:-
un-4un1+3un-2=5n+n