Check whether the following statements are true or not. Justify your a

Question

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) a_n+3a_n-1+2a_n-2= 2" is a linear homogeneous recurrence relation.

iii) A particular solution of the recurrence relation a_n-2a_n-1 + a_n-2=1 has the form Cn².

iv) There exists a boolean expression in variables x,x, and x, with CNF as

$(x_{1}\vee x_{2})\wedge(x_{2}\vee x_{3})\wedge(x_{3}\vee x_{1}).$

v) If a dice is rolled thrice, then the probability of getting a 6 each time is $\frac{1}{72}$

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

vii) Every Eulerian graph is Hamiltonian.

viii) $S_{3}^{4}$ 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.

18 Feb 2025

Answer :

Word Count : 896

Let's go through each statement and determine whether it is true or false, justifying the answers with a short proof or a counterexample. --- ### (i) If the contrapositive of a statement is true, then the statement itself is also true. - A statement of the form $ P \Rightarrow Q $ has a contrapositive $ \neg Q \Rightarrow \neg P $. - The contrapositive of a true statement is always true, and vice versa. - Verdict: True ✅ --- ### (ii) $ a_n + 3a_{n-1} + 2a_{n-2} = 2 $ is a linear homogeneous recurrence relation. - A linear homogeneous recurrence relation has the form: \[ c_k a_n + c_{k-1} a_{n-1} + \dots + c_1 a_1 + c_0 a_0 = 0 \] - Here, the right-hand side is not zero ($2 \neq 0$), so it is not homogeneous. - Verdict: False ❌ --- ### (iii) A particular solution of the recurrence relation $ a_n - 2a_{n-1} + a_{n-2} = 1 $ has the form $ Cn^2 $. - The corresponding homogeneous recurrence is \( a_n - 2a_{n-1} + ___ ____ ___ _______ ___ ____ ____ ________ ________.
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