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Arithmetic Aptitude :: Sets, Relations and Functions

Sets, Relations and Functions - Section-1
  1.  Let f : X → Y . Consider the statement, “For all subsets C and D of Y , f −1 (C∩Dc ) = f −1 (C) ∩ [f −1 (D)]c . This statement is

    2. A.

    True and equivalent to:
    For all subsets C and D of Y , f −1 (C − D) = f −1 (C) − f −1 (D).
    B.

    False and equivalent to:
    For all subsets C and D of Y , f −1 (C − D) = f −1 (C) − f −1 (D).
    C.

    True and equivalent to:
    For all subsets C and D of Y , f −1 (C − D) = f −1 (C) − [f −1 (D)]c
    D.

    False and equivalent to:
    For all subsets C and D of Y , f −1 (C − D) = f −1 (C) − [f −1 (D)]c .

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  2.  The number of partitions of {1, 2, 3, 4, 5} into three blocks is S(5, 3) = 25. The total number of functions f : {1, 2, 3, 4, 5} → {1, 2, 3, 4} with |Image(f)| = 3 is

    3. A.

    4 × 6
    B.

    4 × 25
    C.

    25 × 6
    D.

    4 × 25 × 6

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  3.  Let f : X → Y and g : Y → Z. Let h = g ◦ f : X → Z. Suppose g is one-to-one and onto. Which of the following is FALSE?

    4. A.

    If f is one-to-one then h is one-to-one and onto
    B.

    If f is not onto then h is not onto
    C.

    If f is not one-to-one then h is not one-to-one
    D.

    If f is one-to-one then h is one-to-one

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  4.  Which of the following statements is FALSE?

    5. A.

    {2, 3, 4} ⊆ A implies that 2 ∈ A and {3, 4} ⊆ A
    B.

    {2, 3, 4} ∈ A and {2, 3} ∈ B implies that {4} ⊆ A − B. 
    C.

    A ∩ B ⊇ {2, 3, 4} implies that {2, 3, 4} ⊆ A and {2, 3, 4} ⊆ B
    D.

    A − B ⊇ {3, 4} and {1, 2} ⊆ B implies that {1, 2, 3, 4} ⊆ A ∪ B

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  5.  Let A = {0, 1} × {0, 1} × {0, 1} and B = {a, b, c} × {a, b, c} × {a, b, c}. Suppose A is listed in lexicographic order based on 0 < 1 and B is listed in lexicographic order based on a < b < c. If A×B

    6. A.

    ((1, 0, 0),(b, a, a),(0, 0, 0))
    B.

    ((1, 0, 0),(a, a, a),(0, 0, 1))
    C.

    ((1, 0, 0),(a, a, a),(1, 0, 0)) 
    D.

    ((1, 0, 0),(a, a, a),(0, 0, 0))

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