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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