Fsc Part 1 Mathematics (Complete Solution)

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Q2

The operation ⊕ as performed on the set {0,1,2,3} is shown in the adjoinig tabel, show that the set is an Abelian group?

⊕ |
0 |
1 |
2 |
3 |

0 |
0 |
1 |
2 |
3 |

1 |
1 |
2 |
3 |
0 |

2 |
2 |
3 |
0 |
1 |

3 |
3 |
0 |
1 |
2 |

Solution:

Suppose G={0,1,2,3}

(i) The given table show that each element of the table is member of G thus closure law holds.

(ii) ⊕ is associative in G

(iii) Table show that 0 is identity element w.r.t ⊕.

(iv) Since 0+0=0, 1+3=0, 2+2=0, 3+1=0

=> 0^{-1}=0, 1^{-1}=3, 2^{-1}=2, 3^{-1}=1

(v) As the table is symmetric w.r.t to the principal diagonal. Hence commutative law holds.