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CUSTOMER_CODE
SMUDE
DIVISION_CODE
SMUDE
EVENT_CODE
APR2016
ASSESSMENT_CODE BC0052_APR2016
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72467
QUESTION_TEXT
Define context-free grammar. Obtain the context free grammar for the
regular expression (011 + 1)*(01)*.
Def: A grammar G is a quadruple G = (VN, VT, Φ, S) where
VN is set of variables or non-terminals
VT is set of terminal symbols
Φ is set of productions
S is the start symbol,
is said to be type 2 grammar or context free grammar (CFG) if all the
productions are of the form
SCHEME OF
EVALUATION
A — > α where α (VN VT)* and A VN. The symbol ^ (indicating NULL string)
can appear on the right hand side of any production.
The language generated from this grammar is called type-2 language or
context free language (CFL).
(4 marks)
Solution: The expression (011 + 1 )*(01 )* is of the form A*B* where A = 001
or 1 and B = 01. The regular expression A*B* means that any number of A's
(possibly none) are followed by any number of B's (possibly none). Any
number of A's (that is, 011 's or 1 's) can be generated using the productions
A→011A | 1A | Λ
Any number of B's (that is, 01's) can be generated using the productions
B → 01B|Λ
(3 marks)
Now, the language generated from the regular expression (011 + 1)*(01)* can
be obtained by concatenating A and B using the production
S → AB
(1 mark)
Therefore, the final grammar G = (VN, VT, Φ, S) where
VN: {S, A, B}
VT: {0, 1}
Φ: S → AB,
A →011A|1a|Λ
B → 01B| Λ
S: start symbol.
(2 marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72471
QUESTION_TEXT
Use the definition of order to show that
Solution: The functions f and g referred to in the definition of Onotation are defined as follows. For all real numbers
and
For all real numbers
SCHEME OF
EVALUATION
Therefore,
where C = 4 and
k = 1.
for all x > 1. Or
Hence,
(10 marks)
for all x > k
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72472
QUESTION_TEXT
Prove that “For any finite set A, the cardinality of the power set of A is
2 raised to a power equal to the cardinality of A”.
Solution:
i. Basis Step: Let A be a set of cardinality n = 0. Then
and
on the other hand,
ii.
Induction Hypothesis: Let n > 0, and suppose that
provided that
SCHEME OF
EVALUATION
iii.
Induction Step: Let A be such that
at least one element a. Let
then
Since n > 0, A contains
By the induction
hypothesis,
Now the power set of A can be divided into two parts, those sets
containing the element a and those sets not containing a. The latter
part is just 2B, and the former part is obtained by introducing into each
member of 2B. Thus
This division, in fact partitions 2A into two disjoint equinumerous parts,
so the cardinality of the whole is twice 2|B|, which, by the induction
hypothesis, is
This completes the proof. (10 marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72473
QUESTION_TEXT
If
generated by G.
then find L(G), the language
Solution: Since
is a production,
. Now, for all
Therefore,
we can write the following:
.
In the above derivation, at every step,
the last step where
is applied.
SCHEME OF
EVALUATION
This implies that
is applied, except in
Therefore,
Now suppose
So we should start the derivation of w with S.
If we are applying
first, then we will get
.
Otherwise, the first production that we need to apply is
.
However, at any stage we can apply
to obtain the terminating
string. Therefore w can be derived in the following form.
for some
Hence
(10 marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
110656
QUESTION_TEXT
That is
Define the definition of basic terms
i.
Simple graph and general graph
ii.
Isolated vertex
iii.
Isomorphism
iv.
Circuit and cycle
v.
Finite and infinite graph
Ans:
SCHEME OF
EVALUATION
(i)
Simple graph: A graph that has neither self loops nor parallel
edges is called a simple graph. Graph containing either parallel edges or
loops is also referred as general graph.
(ii)
Isolated vertex: A vertex having no incident edge is called as
isolated vertex. In otherwords, a vertex v is said to be an isolated vertex
if the degree of v is equal to zero.
(iii)
Isomorphism: Two graphs G and G1 is said to be isomorphic
to each other if there is a one-to-one correspondence between their
vertices and a one-to-one correspondence between their edges such that
the incident relationship must be preserved.
(iv)
Circuit and cycle: A path of length ≥ 1 with no repeated edges
and whose endpoints are equal is called a circuit. A circuit may have
repeated vertices other that the endpoints; a cycle is a circuit with no
other repeated vertices except its endpoints.
Finite and infinite graph: A graph G with a finite number of vertices and a
finite number of edges is called a finite graph. A graph ‘G’ that is not a finite
graph is said to be an infinite graph.
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
110658
Define each of the following with an example.
a.
Subset
b.
Union
c.
Cartesian product
d.
Power set
e.
Equal sets
QUESTION_TEXT
a.
Subset: A is a subset of B if every element of A is also an
element of B.
SCHEME OF
EVALUATION
Ex: IF A={1, 2, 3}, B={1, 2, 3, 4}, then A⊆B
b.
Union: IF A and B are two sets, then the set {x|x∈A or x∈B or
both} is union of A and B.
Ex: If A={1, 2, 3}, B={3, 4, 5} then A∪B={1, 2, 3, 4, 5}
c.
Cartesian product: If S and T are two sets, then the set {(s,
t)|s∈S and t∈T} is called the Cartesian product of S and T.
Ex: If X={a, b}, Y={x, y}, then X×Y={(a, x), (a, y), (b, x), (b,
y)}
d.
Power set: Let A be a set. The set of all subsets of A is called
the power set of A.
Ex: If A={1, 2}, then P(A)={∅, {1}, {2, }, {1, 2}}
e.
Equal sets: Two sets A and B are said to be equal if A is a subset
of B and B is a subset of A.
Ex: A={1, 2}, B={1, 2}, then A=B
(Each definition with example carries 2 marks)