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5 July, 00:07

Let A, B, and C be as in, let D = {3, 2}, and let E = {2, 3, 2}. Determine which of the following are true. Give reasons for your decisions.

a. A = B

b. B = C

c. B = D

d. E = D

e. A ∩ B = B ∩ A

f. A ∪ B = B ∪ A

g. A - B = B - A

h. A ⊕ B = B ⊕ A

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Answers (1)
  1. 5 July, 00:31
    0
    The question is incomplete. Below is the complete question "Let A = {0,2,3}, B = {2,3}, C = {1,5,9}, D={3,2}, and let E={2,3,2}. Determine which of the following are true. Give reasons for your decisions.

    a. A = B

    b. B = C

    c. B = D

    d. E = D

    e. A ∩ B = B ∩ A

    f. A ∪ B = B ∪ A

    g. A - B = B - A

    h. A ⊕ B = B ⊕ A"

    Answer

    a. A = B (false)

    b. B = C (false)

    c. B = D (true)

    d. E = D (true)

    e. A ∩ B = B ∩ A (true)

    f. A ∪ B = B ∪ A (true)

    g. A - B = B - A (true)

    h. A ⊕ B = B ⊕ A" (true)

    Step-by-step explanation:

    A. the notation A=B simply means if set A equals set B from the above we can conclude that set A has 3 elements while set B has only two elements, hence the statement is false.

    B. The notation B=C simply means if set B equals set C from the above we can conclude that the sets are not equal since both sets has a different elements

    C. The notation B=D simply means if set B equals set D from the above we can conclude that the sets are equal since both sets has the same elements irrespective of the arrangement.

    D. The notation D=E simply means if set B equals set D from the above we can conclude that the sets are equal since both sets has the same elements irrespective of the repetition

    e. the set

    AnB={0,2,3}n{2,3}

    AnB={2,3}

    also

    BnA={2,3}n{0,2,3}

    BnA={2,3}

    hence A ∩ B = B ∩ A is true

    f. A ∪ B={0,2,3}u{2,3}

    A ∪ B={0,2,3}

    also

    B ∪ A={2,3}u{0,2,3}

    B ∪ A={0,2,3}

    Hence A ∪ B = B ∪ A

    g. The difference between two sets is the set of values in one but not the other

    Hence

    A-B={0,2,3}-{2,3}

    A-B={0}

    also B - A={2,3}-{0,2,3}

    B - A={0}

    Hence A - B = B - A is true

    h. A ⊕ B is the Symmetric difference is those elements that belong to one set, but not the other, it is express as

    A ⊕ B = (A U B) - (A ∩ B)

    also B ⊕ A = (BU A) - (B ∩ A)

    comparing both

    A ⊕ B=B ⊕ A

    (A U B) - (A ∩ B) = (BU A) - (B ∩ A)

    {0,2,3}-{2,3}={0,2,3}-{2,3}

    {0}={0}

    we can therefore conclude that A ⊕ B = B ⊕ A is true
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