In the June, 2021 Term-End Examination of MTE-13, it was asked to give a direct and an indirect proof of the following statement.

such that a is even and a +b is even, then   b is even."

One student gave an indirect proof as follows:

“Let b be m +1, which is an odd number. We already know a and a +b are even. If we substitute b + m +1 in a + b, it becomes a + m +1, which is an odd number. This contradicts the given statement. Hence b is an even number.”

What is wrong with the above proof ? Also give a correct direct and an indirect proof.

What do you understand by a subdivision of a graph? Is every subdivision of a Hamiltonian graph Hamiltonian? Justify.

Is the complement of the Peterson graph planar? Justify your answer.

Is the complement of the Peterson graph planar? Justify your answer.b) Is the complement of the Peterson graph planar? Justify your answer.

Draw three nonisomorphic induced subgraphs of the following graph, each having the same number of vertices. Justify your choice.

For any statements p,q and r, prove that

List all the onto mappings from the set {a,b,c,d} to {1,2,3,4}. How many onto

mappings are there form {a,b,c,d} to {1,2,3,4,5}?

Solve the recurrence relation:

There are about 77 crore ways to arrange the letters of the word “COMBINATORICS”. Count the exact number of such ways.

Using generating functions find 

Express the following statements in symbolic form.

i) There is a man in the park with blue eyes.

ii) Every blue-eyed man in the park is wearing a red hat.

iii) If a man wears no hat, then he has black eyes

Draw the logic circuit for the Boolean expression

Check whether the following statements are true or not. Justify your answers with a short proof or a counter example

i) If the contrapositive of a statement is true, then the statement itself is also true.

ii)   is a linear homogeneous recurrence relation.

iii) A particular solution of the recurrence relationhas the form 

iv) There exists a boolean expression in variables 

v) If a dice is rolled thrice, then the probability of getting a 6 each time is 

vi) Every odd cycle has the same chromatic and edge chromatic numbers.

vii) Every Eulerian graph is Hamiltonian.

viii) 3 4 S gives the number of ways in which any 3 objects can be placed in any 4 boxes.

ix) There exists a self-complementary planar graph on 5 or more vertices.

x) The number of partitions of 6 is 10.

निम्नलिखित के स्तब्ध बिन्दु निर्धारित करके उनका वर्गीकरण कीजिए :

(i) 

(ii)

क्या निम्नलिखित फलन   पर संतुष्ट करता है? अपने उत्तर की पुष्टि कीजिए। x0, y 0 श्वार्ज-प्रमेय की आवश्यकताओं को (1,1)

निम्नलिखित फलोन्स के स्तर वज्र ज्ञात किओ:

(i) 

(ii) 

(iii) x − y

(iv) y / x

निम्नलिखित में से कौनसा कथन सत्य है? अपने उत्तर की पुष्टि कीजिए।

(i) S ⊂ C

(ii)  C ⊂ S

मान लीजिए कि S और CR³ के उपसमुच्चय हैं। S मूल-बिन्दु पर केन्द्र वाला एकक विवृत गोलक है तथा C विवृत घन  

  यदि तो a और b के मान ज्ञात कीजिए।

द्विशः समाकलन का प्रयोग करके दीर्घवृत्तज  का आयतन ज्ञात कीजिए।