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IGNOU MTE 13 Discrete Mathematics Solved Assignment 2024

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Last Date of Submission of IGNOU MTE-013 (BDP) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU MTE 13 2024 Solution
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BDP
Course Name	Bachelor Degree Programmes
Subject Code	MTE 13
Subject Name	Discrete Mathematics
Year	2024
Session	-
Language	English Medium
Assignment Code	MTE-013/Assignmentt-1//2024
Product Description	Assignment of BDP (Bachelor Degree Programmes) 2024. Latest MTE 013 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-013 (BDP) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MTE 13/2024

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Questions Included in this Help Book

Ques 1.

State whether the following statements are true or false. Justify your answer with a short explanation or a counter-example. (2 × 10 = 20)
(i) The contra- positive of the statement $"x^2+y^2=0\Rightarrow x=0$ and $y=o"$ is $"x\neq 0$ and $"y\neq 0\Rightarrow x^2+y^2\neq 0"."$
(ii) $a^n=\frac{1}{9}[2^{n+1}+(-1)^n]^2$ is the solution of the recurrence relation
$\sqrt{a_n}=\sqrt{a_{n-1}}+2\sqrt{a_{n-2}},(n\geq 1),a_0=a_1=1.$
(iii) The number of integers between 1 and 360 which are relatively prime to 360 are
96.[ Note that 1 is relatively prime to every positive integer.]
(iv) The coefficient of $x^6y^5z^9$ in $(x+y+z)^{20}$ is $C(20,6).C(20,5).C(20,9).$
(v) If a graph has 6 vertices and 10 edges, then it can’t be regular.
(vi) Peterson graph is bipartite.
(vii) Every edge of a Tree is a bridge.
(viii) The independence number of $C_5,i.e.,\alpha (C_5)$ is $2.$
(ix) The DNF of the expression $p(x,y,z)=(x\wedge z)^{'}$ is $(x\vee y\,\vee z^{'})\vee(x\,\vee\,y^{'}\vee\,z^{'}).$
(x) The number 8 has atmost one self-conjugate partition.

Ques 2.

Find the number of integer solutions of the equation $x_1+x_2+x_3+x_4=0,x_1 \geq -4,\forall \,i$ using the generating function technique.

Ques 3.

Let $a_n$ be the number of

Ques 4.

Solve the recurrence relation $a_n=2a_{n-1}+n(n-1),n\geq 1,a_0=1,$ using the generating function technique.

Ques 5.

Look at the following pattern:

(i) How many ′ + ′ signs should be there in the box?

(ii) What are the smallest and largest integers in the box?

Ques 6.

Construct the logic circuit of the Boolean expression $(x^{'}_1\vee \,x^{'}_2\,\wedge x_3)\vee(x_2\,\wedge\,x^{'}_3 ),$ where $x_1,x_2,x_3$ are the inputs in that circuitry.

Ques 7.

.Solve the recurrence relation $a_{n+1}=a_n+n.2^n,(n\geq0),a_0=1,$ using the method of telescopic sums.

Ques 8.

Prove by contradiction that the inverse of a square matrix, if exists, is unique.

Ques 9.

Give a direct proof of the following statement: “Every skew-symmetric matrix of odd order is singular”.

Ques 10.

For any propositions

Ques 11.

Let $p,q$ and $r$ be the statements as defined below:

$p$ ∶ The Course is enjoyable.
q ∶ The presentation is stimulating.
r ∶ The material is significant.

Write each of the following in symbolic form.

(i) The material is significant and the presentation is stimulating, but the course is not enjoyable.
(ii) It is not the case that both the course is enjoyable and, at the same time,the presentation is not stimulating.

Ques 12.

Find out which of the following are propositions and which are not. Give reasons for your answers.

(i) There are a total of 8 questions in this assignment.
(ii) Wow, it’s a beautiful evening!
(iii) I am not sure whether I can run as fast as you can.
(iv) The set {2, −1, 2, −1, 2, −1, … } is unbounded.
(v) The set ℕ of natural numbers is equivalent to the set ℤ of integers.
(vi) Rajasthan is the largest state in area among the Indian states and union territories.

Ques 13.

Using the principle of mathematical induction show that the sum of the cubes of three successive positive integers is divisible by 9.

Ques 14.

Obtain the CNF of the Boolean expression $X(x_1,x_2,x_3)=x_2\,\vee(x_1\,\wedge x^{'}_3).$

Ques 15.

A Cantabil showroom offers 7 styles of pants. For each style, there are 10 different possible waist sizes, 6 pants lengths and 4 color choices. How many different types of pants could the showroom have?

Ques 16.

Find the number of nonnegative integer solutions of the equation $x_1+x_2+x_3+x_4+x_5=12$ . And how many solutions in positive integers will it have? Justify.

Ques 17.

What is the probability that an integer between 1 and 9,999 has exactly one 8 and one 9 ?Justify your answer.

Ques 18.

Check wether the graph in Fig. 1 is Eulerian or not. Is it regular? Justify your answer.

Ques 19.

Check whether the graphs in Fig. 2 are planar or not. If yes, give their plane drawing, otherwise give a $K_{3,3}$ or $K_{5}$ subdivision.

Ques 20.

Using the method of generating functions, solve the recurrence relation $a_{n}=2a_{n-1}+2^{n},~~n\geq1~,a_{0}=1.$

Ques 21.

Using combinatorial arguments, prove that $(a+b+c)^{n}=\Sigma_{r+s+t=n}\left(\!\!\!\begin{array}{c}{{n}}\\ {{r\,,\,\,s,\,\,t}}\end{array}\!\!\right)a^{r}b^{s}c^{t}.$ What is the coefficient of $x^{50}y^{12}z^{38}\stackrel{.}{\mathrm{in}}(x-y+2z)^{100}?$

Ques 22.

Let

Ques 23.

How will you define a complete bipartite graph? Give an example of it. Is every complete bipartite graph complete? Justify.

Ques 24.

Using combinatorial arguments, prove tha $\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r},$ where the notation $\binom{n}{k}$ is used for the number of ways to choose a subset of

Ques 25.

Write down the degree sequences of the graphs in Fig. 3:

Are these graphs isomorphic? Justify your answer.

Ques 26.

Can the following figures be drawn without lifting the pen from paper and without covering any line segment more than once? Give reasons in support of your answers.

Ques 27.

How many colors are needed to color the 15 balls in the below given triangular array so that no two touching balls get same color?

Ques 28.

Prove that the graph given below is non-Hamiltonian.

Ques 29.

What is the edge-chromatic number of the graph given in Fig. 3(ii)? Justify by giving an explicit coloring

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