Sets Class Problems

1. Consider the following sets:
A = {x∈: there is n∈ such that x = 2n+1}
B = {x∈: there is n∈ such that x = 4n+1}
C = {x∈: there is n∈ such that x = 3n+3}
D = {x∈: there is n∈ such that x = 3n}
Determine the truth value of the following statements:

◼
  • We can start by writing the first few terms of our sets:
    • ◼
  • A={1,3,5,7,9,...}
    • ◼
  • B= {1,5,9,...}
    • ◼
  • C= {3,6,9,12,...}
    • ◼
  • D={0,3,6,9,12,...}
    • 1
    )
    4 ∈ A
    false
    2
    )
    3 ∈ C
    true
    3
    )
    5 ∈ A⋂B
    true
    4
    )
    A ⊆ B
    false
    5
    )
    B ⊆ A
    true
    6
    )
    C ⊆ D
    true
    7
    )
    D ⊆ C
    false
    8
    )
    0 ∈ C
    false

    2. Consider the following sets:
    A = {1,2,3,4,5}
    B = {2,4,6,8,10,12}
    C = {3,6,9,12}

    1
    )
    Determine the following values:
    1
    .
    1
    .
    |A|
    5
    1
    .
    2
    .
    |B|
    6
    1
    .
    3
    .
    |C|
    4
    1
    .
    4
    .
    |A⋂B|
    A⋂B={2,4}
    2
    1
    .
    5
    .
    |A⋂C|
    A⋂C= {3}
    1
    1
    .
    6
    .
    |B⋂C|
    B⋂C={6,12}
    2
    1
    .
    7
    .
    |A⋃B|
    A⋃B={1,2,3,4,5,6,8,10,12}
    9
    1
    .
    8
    .
    |A⋃C|
    A⋃C = {1,2,3,4,5,6,9,12}
    8
    1
    .
    9
    .
    |B⋃C|
    B⋃C={2,3,4,6,8,9,10,12}
    8
    2
    )
    For sets X and Y determine the equality relation between |X|,|Y|, |X⋂Y|, and |X⋃Y|
    |X⋃Y| = |X| +|Y| -|X⋂Y|

    3. Write the following sets explicitly

    1
    )
    {x∈: x<100 and x is spelled in alphabetical order in English}.
    {40}
    2
    )
    Power set of {a,b,c}.
    {{},{a},{b},{c},{a,b},{b,c},{c,a},{a,b,c}}
    3
    )
    The set of subsets of {1,2,3,4} with 2 elements.
    {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

    4. For arbitrary sets X and Y, determine the truth value of the following statements. If the statement is true, show it by using the definitions, if it is false provide a counterexample.

    1
    )
    X⋃Y = Y⋃X
    True:
    X ⋃ Y = {x ∈  : x ∈ X or x ∈ Y} = {x ∈  : x ∈ Y or x ∈ X} = Y ⋃ X
    2
    )
    X⋂Y = Y⋂X
    True
    X ⋂ Y = {x ∈  : x ∈ X and x ∈ Y} = {x ∈  : x ∈ Y and x ∈ X} = Y ⋂ X
    3
    )
    X\Y = Y \X
    False
    Let X = {1,2}, Y = {2,3}. Then X\Y = {1}, and Y\X = {3} which are different.
    4
    )
    XY = YX
    False
    Let X = {1,2} and Y = {a,b}. Then
    XY = {(1,a), (1,b), (2,a), (2,b)}
    YX = {(a,1), (a,2), (b,1), (b,2)} notice that the tuple (1,a) is different than the tuple (a,1) so the elements of XY and YX are not the same.

    5. Let a, b ∈ . What are the possible values for |{a,b}|?

    When a = 1, and b = 2, then | {1, 2} | = 2
    a = 1, b =1 |{1,1}|=|{1}|=1

    6. Let n∈, and A a set with n distinct elements. Answer the following questions. (Start with a few cases for low values of n if you need help)

    1
    )
    How many subsets does A have?
    0->1
    1->2
    2->{{},{1},{2},{1,2}}->4
    3->8
    4->16
    ​
    n
    2
    2
    )
    How many subsets with cardinality 1 does A have?
    0 -> 0
    1 -> 1
    2 -> 2
    3 -> 3
    n -> n
    3
    )
    How many subsets with cardinality n does A have?
    This is the same as the set A itself, so just 1.
    4
    )
    How many subsets with cardinality n-1 does A have?
    n. Because this is the same as choosing which element of A is not in the subset.

    7. Consider the definition for A = B when A and B are sets.

    1
    )
    Write the definition
    That is, A=B if and only if every a∈A is an element of B and every b∈B is an element of A
    2
    )
    Using the definition above, explain why order of the elements does not matter.
    A={1,2},B={2,1}
    To check A equals B we take each of the elements in A and check if they belong to B and vice-versa
    3
    )
    Using the definition, explain why multiple occurrences of the same element don’t matter.
    A={1,2},B={2,1,1,1}

    8.Let A = {1,2,3,4}, B = {3,4,5,6}, ={1,2,3,4,5,6,7,8,9,10}.

    1
    )
    Write explicitly the set C={x∈ : x∈A x∈B}
    {3,4,5,6,7,8,9,10}
    2
    )
    Write Explicitly the set D = {x∈: x∈A ⧦x∉B}
    {1,2,5,6}

    9. Determine the truth value of the following.

    1
    )
    For all sets A, B. we have either A⊆B, B⊆A, or A=B
    False
    A={1,2},B={2,3}