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### Arithmetic Aptitude :: Algebra Problems

1.  The set of all real numbers under the usual multiplication operation is not a group since

2.  A. zero has no inverse B. identity element does not exist C. multiplication is not associative D. multiplication is not a binary operation

3.  If (G, .) is a group such that (ab)- 1 = a-1b-1, ∀ a, b ∈ G, then G is a/an

4.  A. commutative semi group B. non-abelian group C. abelian group D. None of these

5.  If (G, .) is a group such that a2 = e, ∀a ∈ G, then G is

6.  A. abelian group B. non-abelian group C. semi group D. none of these

7.  The inverse of - i in the multiplicative group, {1, - 1, i , - i} is

8.  A. -1 B. 1 C. -i D. i

9.  The set of integers Z with the binary operation "*" defined as a*b =a +b+ 1 for a, b ∈ Z, is a group. The identity element of this group is

10.  A. -1 B. 0 C. 1 D. 2

11.  If R = {(1, 2),(2, 3),(3, 3)} be a relation defined on A= {1, 2, 3} then R . R (= R2) is

12.  A. {(1, 2),(1, 3),(3, 3)} B. {(1, 3),(2, 3),(3, 3)} C. {(2, 1),(1, 3),(2, 3)} D. R itself

13.  Which of the following statements is false ?

14.  A. If R is relexive, then R ∩ R-1 ≠ φ B. R ∩ R-1 ≠ φ   =>R  is anti-symmetric. C. If R, R' are relexive relations in A, then R - R' is reflexive D. If R, R' are equivalence relations in a set A, then  R ∩ R'  is also an equivalence relation in A

15.  If (G, .) is a group, such that (ab)2 = a2 b2 ∀ a, b ∈ G, then G is a/an

16.  A. abelian group B. non-abelian group C. commutative semi group D. none of these

17.  (Z,*) is a group with a*b = a+b+1 ∀ a, b ∈Z. The inverse of a is

18.  A. 0 B. -2 C. a-2 D. -a-2

19.  Let G denoted the set of all n x n non-singular matrices with rational numbers as entries. Then under multiplication G is a/an

20.  A. subgroup B. ininite, abelian C. finite abelian group D. infinite, non abelian group