three

Relationship matrix M=

0 1 0 0

1 0 1 0

0 0 0 1

0 0 0 0

R = { & lta，b & gt，& ltb，a & gt，& ltb，c & gt，& ltc，d & gt}

Reflexive closure r (r) =

1 1 0 0

1 1 1 0

0 0 1 1

0 0 0 1

Symmetric closure s (r) =

0 1 0 0

1 0 1 0

0 1 0 1

0 0 1 0

Transitive closure t (r) =

1 1 1 1

1 1 1 1

0 0 0 1

0 0 0 0

four

calculate

0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 0

2 2 3 4 0 1

3 3 4 0 1 2

4 4 0 1 2 3

* Operation

* 0 1 2 3 4

0 0 0 0 0 0

1 0 1 2 3 4

2 0 2 4 1 3

3 0 3 1 4 2

4 0 4 3 2 1

five

The group unit is 1.

Any power of 0 and 5 (0 or 5 can be added at any time) is 0, so the unit element 1 cannot be obtained.

So the order of 0 and 5 is infinite.

1? = 1, then the order of 1 is 1.

2+2+2 ≡ 1(mod 5) So the order of 2 is 3.

3+3 ≡ 1(mod 5) So the order of 3 is 2.

4+4+4+4 ≡ 1(mod 5) So the order of 4 is 4.