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.

i) TRUE
If the contrapositive of a statement is true, then the original statement is logically equivalent and must also be true. (Contrapositive of "if P then Q" is "if not Q then not P".)

ii) FALSE
The statement is incomplete. To be a linear homogeneous recurrence relation, it must be of the form:
aₙ + c₁aₙ₋₁ + c₂aₙ₋₂ + ... + cₖaₙ₋ₖ = 0.
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