Question No: 41 ( Marks: 2 )
Find a non-isomorphic tree with five vertices.
There are three non-isomorphic trees with five vertices as shown
(where every tree with five vertices has 5-1=4 edges).
Question No: 42 ( Marks: 2 )
Define a predicate.
A predicate is a sentence that contains a finite number of
variables and becomes a statement when specific values are
substituted for the variables.
The domain of a predicate variable is the set of all values that
may be substituted in place of the variable.
Let the declarative statement:
“x is greater than 3”.
We denote this declarative statement by P(x) where
x is the variable,
P is the predicate “is greater than 3”.
The declarative statement P(x) is said to be the value of the
propositional function P at x.
Question No: 43 ( Marks: 2 )
Write the following in the factorial form:
(n +2)(n+1) n
( 2)( 1) !
!
n n n
n
+ +
Question No: 44 ( Marks: 3 )
Determine the probability of the given event
“An odd number appears in the toss of a fair die”
Sample space will be..S={1,2.3,4,5,6}…there are 3 odd numbers so,
For odd numbers, probability will be
3
6…Ans
Question No: 45 ( Marks: 3 )
Determine whether the following graph has Hamiltonian circuit.
This graph is not a Hamiltonian circuit, because it does not
satisfy all conditions of it.
E.g. it has unequal number of vertices and edges. And its path
cannot be formed without repeating vertices.
Question No: 46 ( Marks: 3 )
Prove that If the sum of any two integers is even, then so is their
difference.
Theorem: ∀ integers m and n, if m + n is even, then so is m - n.
Proof:
Suppose m and n are integers so that m + n is even. By definition
of even, m + n = 2k for some integer k. Subtracting n from both
sides gives m = 2k - n. Thus,
m - n = (2k - n) -
n
by
substitution
= 2k - 2n
combining
common
terms
= 2(k - n)
by
factoring
out a 2
But (k - n) is an integer because it is a difference of integers.
Hence, (m - n) equals 2 times an integer, and so by definition of
even number, (m - n) is even.
This completes the proof.
Question No: 47 ( Marks: 5 )
Show that if seven colors are used to paint 50 heavy bikes, at
least 8 heavy bikes will be the same color.
N=50
K=7
C(7+50-1,7)
C(56,7)
56!/(56-7)!7!
56!/49!.7!
Question No: 48 ( Marks: 5 )
Determine whether the given graph has a Hamilton
does, find such a circuit, if it does not , given an argument to
show why no such circuit exists.
(a)
This graph does not have Hamiltonian circuit, because it does not
satisfy the conditions. Circuit may not be completed without
repeating edges. It has also unequal values of edges and vertices.
(b)
This graph is a Hamiltonian circuit ..Its path is a b c d e a
Question No: 49 ( Marks: 5 )
Find the GCD of 11425 , 450 using Division Algorithm.
LCM = 205650
11425 = 450x25 + 175
450 = 175x2 + 100
175 = 100x1 + 75
100 = 75x1 + 25
75 = 25x3 + 0
Linear combination= 25 = 127x450 + -5x11425
GCD= 25…Ans
Question No: 50 ( Marks: 10 )
Write the adjacency matrix of the given graph also find
transpose and product of adjacency matrix and its transpose
(if not possible then give reason)
Adjacency matrix= 0 1 0 0 0
1 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 0
Transpose = 0 1 0 0 0
1 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 0
Its transpose is not possible…it’s same. Because there is no loop.
It is not directed graph.
By ADEEL ABBAS, Bhakkar. AdeelAbbasbk@gmail.com
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