IGNOU MCS 212 Exam Notes / Important Question

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Verify whether √11 is rational or irrational
What are Demorgan’s Law? Explain the use of Demorgen’s law with example.
Make truth table for followings:
i) p→ (~ q ∨ ~ r) ∧ (p ∨ r)
ii) p→(~ r ∧ q) ∧ (p ∧ ~ q)
Obtain the truth value of disjunction of “ Water is essential for life” and “2+2=4”.
Write the following statements in the symbolic form.
i) Some students can not appear in exam.
ii) Everyone can not sing.
Draw logic circuit for the following Boolean Expression:
(x y z) + (x+y+z)’+(x’zy’ )
What is dual of a boolean expression? Explain with the help of an example.
Show using truth table whether (P ∧ Q ∨ R) and (P ∨ R) ∧ (Q ∨ R) are equivalent or not.
Explain whether (P ∧ Q) → (Q → R) is a tautology or not
What is power set? Write power set of set A={1,2,3,4,5,6,7,9}
Give geometric representation for followings:
i) { -3} x R
ii) {1, -2) x ( 2, -3)
What is proper subset? Explain with the help of example.
What is relation? Explain properties of relations with example.
Write the finite automata corresponding to the regular expression (a + b)*ab
If L1 and L2 are context free languages then, prove that L1 U L2 is a context free language.
Explain Decidable and Undecidable Problems. Give example for each
What is equivalence relation? Explain use of equivalence relation with the help of an example.
Suppose we want to choose two persons from a party consisting of 35 members as Manager and Assistant Manager. In how many ways can this be done?
There are three Companies, C1, C2 and C3. The party C1 has 4 members, C2 has 5 members and C3 has 6 members in an assembly. Suppose we want to select two persons, both from the same Company, to become president and vice president. In how many ways can this be done?
Suppose there are five married couples and they (10 people) are made to sit about a round table so that neither two men nor two women sit together. Find the number of such circular arrangements
How many words can be formed using letter of DEPARTMENT using each letter at most once?
i) If each letter must be used,
ii) If some or all the letters may be omitted.
What is the probability that a 13-card hand has at least one card in each suit?
What is the probability that a number between 1 and 10,000 is divisible by neither 2, 3, 5 nor 7?
Explain inclusion-exclusion principle and Pigeon Hole Principle with example.
In a tennis tournament, each entrant plays a match in the first round. Next, all winners from the first round play a second-round match. Winners continue to move on to the next round, until finally only one player is left as the tournament winner. Assuming that tournaments always involve n = 2k players, for some k, find the recurrence relation for the number rounds in a tournaments of n players.
Find an explicit recurrence relation for minimum number of moves in which the n-disks in tower of Hanoi puzzle can be solved! Also solve the obtained recurrence relation through an iterative method.
Draw 2-isomorphic graphs and 3 non- isomorphic graphs on five vertices
Draw the following graphs and state which of following graph is a regular graph?
(i)C5 (ii) W5 (iii) Q4 (iv) K5,5
Determine whether the above graph has a Hamiltonian circuit. If it has, find such a circuit. If it does not have, justify it
Explain and prove the Handshaking Theorem, with suitable example
Show that C6 is bipartite and K3 is not bipartite.
Explain the terms PATH, CIRCUIT and CYCLES in context of Graphs.
What is the difference between an Eulerian graph and an Eulerian circuit?
Explain the Dirac’s Criterion and Ore’s Criterion for Hamiltonian graphs
What are laws related to equivalence propositions ? Give example for each of equivalence proposition laws.
What are modus ponens and modus tollens? Give example for each.
Show that the number √11 is irrational.
Define set, power set, subset, superset and proper set with suitable examples.
Differentiate between symmetric relation and transitive relation with example.
Using the properties pf closure for any set s of strings prove that s* = (s* ) * = s**
What is regular expression? How it differs from regular language?
What is a Turing Machine ? Define Turing Acceptable Language and Turing Decidable Language with example
Rohan rolled two dice red and blue. Calculate the probability of Rohan getting a big number on red dice than the number on blue dice.
Calculate the total number of words that can be formed using the letters of the word “MISSIPPI” if two ‘S’ and two ‘I’ are adjacent to each other.
State and prove Pascal’s Formula w.r.t binomial coefficients.
State and prove Pigeonhole principle.
Explain the application of inclusion-exclusion to Surjective Functions with example.
In how many ways can 30 students be grouped into 7 groups?
Differentiate between path, walk, circuit and cycle in a graph with example
Differentiate between Eulerian and Hamiltonian graphs with suitable examples.
Write down the mathematical notations for the following:
(a) The set of all even numbers.
(b) The set of all natural number whose square is more than 21.
Define the following terms using mathematical notations. Also provide example for each term
 Subset
 Universal set
 Power set
Show the following operations on set(s)
 Complement of a set
 Symmetric difference of two sets
What are the two types of indirect proofs? Explain through an example for each type.
Consider the simple problem of placing four coloured balls: red, blue, green and white in 15 boxes. What are the numbers of distinct ways in which the balls can be placed in these boxes, if each box can hold only one ball? Also write the generalized formula of this numerical result.
Define K-edge colouring and K-vertex colouring of a graph. Find the edge chromatic numbers to colour the edges of the complete graph with four and five vertices.
Describe the following properties of a binary relation with the help of examples.
 Antisymmetry
 Transitivity
What is the difference between Eularian graph and Eularian circuit?
Write a proof for the following theorem:
Theorem- A connected graph is Eularian if and only if the degree of each of its vertices is even

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