Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 8H / b

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- Solution b

Solution

Reading the directed edges with positive transition probability, the communicating classes are

{3}, {4}, {1, 2, 5, 6} .

State

3

can move to

2

or

4

, and state

4

can move to

2

or

5

, so their singleton classes are not closed. Every transition from

1, 2, 5,

or

6

remains within

{1, 2, 5, 6}

, so

{1, 2, 5, 6} is the unique closed class .

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