STUPIDSID
1(a)
Find how many integers between 1 and 60 are not divisible by 2 by nor by 3 and nor by 5 repectively.
6 M
1(b)
By using mathematical induction prove that \(1+a+a^2+....a^n = \frac{1-a^{n+1}}{1-a} \)/, where n≥0
8 M
1(c)
Let A={1,
2,
3,
4,
5} and R be the relation defined by a R b if and only if a R2 and R3. Draw digraph of R,
R2 and R3.
2,
3,
4,
5} and R be the relation defined by a R b if and only if a
R2 and R3.
8 M
2(a)
Show that a group G is Ablian, if and only \( \left ( ab \right )^2 = a^2b^2 \)/ for all elements a and b in G.
6 M
2(b)
Let A={1,
2,
3,
4,
6}=B,
a R b if and only if a is multiple of b Find R. Find each of the Following (i) R(4)
ii) R(G)
iii) R{2,
4,
6}).
2,
3,
4,
6}=B,
a R b if and only if a is multiple of b Find R. Find each of the Following (i) R(4)
ii) R(G)
iii) R{2,
4,
6}).
6 M
2(c)
Show that the (2,
5) encoding function e: B2→B5 defined by e(00) =00000
e(01) = 01110
e(10) =10101
e(11) = 11011 is a group code. How many errors will it detect and correct?
5) encoding function e: B2→B5 defined by e(00) =00000
e(01) = 01110
e(10) =10101
e(11) = 11011 is a group code. How many errors will it detect and correct?
8 M
3(a)
State pigeon hole and extended pigeon hole principle. Show that 7 colors are used to paint 50 bicycles, at least 8 bicycle will be of same color.
6 M
3(b)
Define distributive lattice. Show that in a bounded distributive lattice, if a complement exists, its unique.
6 M
3(c)
Functions f,
g,
h are defined on a set, X={1,
2,
3} as f={(1,
2)
(2,
3)
(3,
1)} g = { (1,
2}
(2,
1)
{3,
3)} h = {(1,
1)
(2,
2)
(3,
1) }
i) Find f o g,
g o f are they equal?
ii) Find f o g o h and f o h o g
g,
h are defined on a set, X={1,
2,
3} as f={(1,
2)
(2,
3)
(3,
1)} g = { (1,
2}
(2,
1)
{3,
3)} h = {(1,
1)
(2,
2)
(3,
1) }
i) Find f o g,
g o f are they equal?
ii) Find f o g o h and f o h o g
8 M
4(a)
Define Eular path and Eular circuit, determine whether the given graph has Eular path and Eular circuit.
!mage
!mage
6 M
4(b)
Define Hamiltonian path and Hamiltonian circuit, determine whether the given graph has Hamiltonian path and Hamiltonian circuit.
!mage
!mage
6 M
4(c)
Define isomorphic graphs. Show that the following two graphs are isomorphic.
!mage
!mage
8 M
5(a)
What is an Universal and existential quantifiers? Prove that distribution law.(p∨q∧r)=(p∨q)∧(p∨r)
6 M
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