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    Set Theory questions

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    1. Let T = {m   m = 1 + (-1)i, for some integer i}. Describe T.
    2. Let A = {m   m = 2i - 1, for some integer i},
    B = {n   n = 3j + 2, for some integer j},
    C = {p   p = 2r + 1, for some integer r}, and
    D = {q   q = 3s - 1, for some integer s}.
    a) Describe the 4 sets (by enumerating their elements).
    b) Is A = D? Explain.
    c) Is B = D? Explain.
    3. Let R, S, and T be defined as follows:
    R = {x    x is divisible by 2},
    S = {y    y is divisible by 3},
    T = {z    z is divisible by 6}.

    a) Describe the 3 sets (by enumerating their elements).
    b) Is T  S? Explain.
    c) Find R  S. Explain.
    4. Let A = {a, b, c}, B = {b, c, d}, and C = {b, c, e}.
    a) Find A  ( B  C), (A  B)  C, and (A  B)  (A  C). Which of these sets are equal?
    b) Find (A  B)  C and A  (B  C). Are these sets equal?
    5. For all sets A, B, and C,
    (A  B)  (C  B) = A  (B  C).
    Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
    6. For all sets A, B, and C,
    if A  C  B  C and A  C  B  C, then A = B.
    Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
    7. For all sets A, B, and C,
    (A  B)  C = A  (B  C).
    Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
    8. Derive the following property:
    For all sets A, B, and C,
    (A  B)  C = (A  C)  B.
    9. Derive the following property:
    For all sets A and B,
    A  (A  B) = A  B.
    10. Suppose A, B, and C are sets.
    a) Are A  B and B  C necessarily disjoint? Explain.
    b) Are A  B and C  B necessarily disjoint? Explain
    c) Are A  (B  C) and B  (A  C) necessarily disjoint? Explain.
    11. Let S = {a, b, c} and for each integer i = 0, 1, 2, 3, let Si be the set of all subsets of S that have i elements. List the elements in S0, S1, S2, and S3. Is {S0, S1, S2, S3} a partition of (S) (the power set of S)?

    © BrainMass Inc. brainmass.com October 10, 2019, 8:22 am ad1c9bdddf
    https://brainmass.com/math/recurrence-relation/set-theory-questions-621911

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    1. Let T = {m   m = 1 + (-1)i, for some integer i}. Describe T.

    Solution:

    When i = 0
    m = 1 + (-1)0 = 1+1 = 2

    When i is odd integer
    m = 1+(-1) = 0

    When i is even integer
    m = 1+1 = 2

    Thus, T = {0, 2}

    2. Let A = {m   m = 2i - 1, for some integer i},
    B = {n   n = 3j + 2, for some integer j},
    C = {p   p = 2r + 1, for some integer r}, and
    D = {q   q = 3s - 1, for some integer s}.
    a) Describe the 4 sets (by enumerating their elements).

    Solution:

    A = {--------------------- -7, -5, -3, -1, 1, 3, 5, 7,...................}
    B = {----------------------7, -4, -1, 2, 5, 8, 11,.......................}
    C = {--------------------- -7, -5, -3, -1, 1, 3, 5, 7,...................}
    D = {----------------------7, -4, -1, 2, 5, 8, 11,.......................}

    b) Is A = D? Explain.

    Solution:

    From part (a), we can note that A≠D as the elements in the sets A and D are not the same.

    Answer: NO

    c) Is B = D? Explain.

    Solution:
    From part (a), we can note that B=D as the elements in the sets B and D are the same.

    Another proof:
    Let n B
    Then n = 3j + 2
    We can write it as
    n = 3j+3-1 = 3(j+1)-1
    let s = j+1
    Thus, n = 3s-1, where s is some integer

    This means n D
    This gives ---------------------(i)

    Now, let m D
    Then m = 3s-1
    We can write it as
    m = 3s-3+2 = 3(s-1)+2
    Let j = s-1
    Thus, m = 3j+2

    This means m B
    This gives ---------------------(ii)

    Now, from (i) and (ii)
    B = D

    Answer: YES

    3. Let R, S, and T be defined as follows:
    R = {x    x is divisible by 2},
    S = {y    y is divisible by 3},
    T = {z    z is ...

    Solution Summary

    This posting explains how to prove or disprove statements using counterexamples in set theory.

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