**Roll No………**

**Total No. of Questions: 9**

**Paper ID [B0104]**

**MCA (Sem.-1**

^{st})**COMPUTER MATHEMATICAL FOUNDATION**

**SUBJECT CODE: (MCA-104)**

**Time: 3 Hrs. Max. Marks: 60**

**Instruction to Candidates:**

1. Attempt any One question from each Section-A, B, C, & D.

2. Section-E is Compulsory.

3. Use of non-programmable Scientific Calculator is allowed.

**SECTION-A**

Q1. (a) Let A and B be sets, then (A

(b)
A town has population of 60,000. Out it 32, 000 read ‘The Hindustan Times’
paper and 35,000 read ‘Times of India’ paper, while 7,500 both the newspapers.
Indicate how many read neither The Hindustan Times not Times of India.

Q2.
Prove that every equivalence relation
on a set generates a unique partition of the set. The blocks of this partition correspond to
the R-equivalence classes.

**SECTION-B**

Q3. (a) Prove by using
induction that if n then 1.11 +2.2! +….. +n. nt=(n+1)!-1.

(b)
Prove (P ^ q) – P v q is a tautology but ( p v q) – (p^ q) is not.

Q4.
(a) Let M(x) be “x is a mammal”. Let A(x) be “x is an animal” and let W(x) be
“x is warm blooded.”

(i)
Translate into formula: every mammal is warm blooded.

(ii)
Translate into English: (3x) [A(x) ^ ~M(x)].

(b)
Prove by construction of truth tables that

~ (p-q)= ~(~P v q)= p ^ ~q.

**SECTION-C**

Q5. Solve the following system of equations by Gauss elimination
method:

X

_{1}+2x_{2}+3x_{3}+4x_{4}=10
7x

_{1}+10x_{2}+5x_{3}+2x_{4}=40
13x

_{1}+6x_{2}+2x_{3}-3x_{4}=34
11x

_{1}+14x_{2}+8x_{3}-x_{4}=64
Q6. (a) Find the inverse of the matrix A= Also determine its rank.

(b)
Prove that inverse of non-singular
symmetric matrix is symmetric.

**SECTION-D**

Q7. (a) Define Euler graph.
Prove that a connected graph G is an Euler graph if and only if all the
vertices of G are even degree.

(b)
Define a bipartite graph. Prove that a graph is bipartite if and only if it
contains no circuit of odd lengths.

Q8. (a)Define planar graph. Prove that the graph k

_{5}is not planar.
(b)
Describe an algorithm for finding shortest path.

**SECTION-E**

Q9. (a) What do you mean by equivalent sets?

(b)
State De-Morgan’s law.

(c) Define
complement of a set.

(d)What
is meant by domain and range of a relation?

(e) Let
X={1,2}, R={(1,1), (2,1),(2,2)}, S={(1,2),(2,1),(2,2)}, Verify that (SoR)

^{-1}.
(f)
Construct a truth table for P ^ ~ P.

(g) Use
universal quantifiers to state that the sum of two rational numbers is
rational.

(h) What
do you understand by the term rank of a matrix?

(i) Does
the inverse of a square matrix always exist? If yes, prove the statement and if
not state the condition under which the inverse of square matrix exists.

(j) What
is a simple graph? Give an example.

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