Set theory : Check whether (A U B )^c = A^c U b^c ?

Let U = N

A = [ n ∈ N , 3 divides n ]

B = [ n ∈ N , 5 divides n ]

Check whether (A U B)^c = A^c U B^c

where c is complement

plz help

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  • Anonymous
    9 years ago
    Favorite Answer

    A U B = { n ∈ N : 3 or 5 divides n }

    (A U B)^c = { n ∈ N : neither 3 nor 5 divides n }

    A^c = { n ∈ N : 3 does not divide n }

    B^c = { n ∈ N : 5 does not divide n }

    A^c U B^c = { n ∈ N : either 3 or 5 does not divide n }

    So the statement (A U B)^c = A^c U B^c is not true.

    6 is an element of A^c U B^c but not of (A U B)^c

    Perhaps you can see this more clearly if I enumerate the first few elements of each set.

    A U B = { 3, 5, 6, 9, 10, 12, 15, ... }

    (A U B)^c = { 1, 2, 4, 7, 8, 11, 13, 14, 16, ... }

    A^c = { 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16 ... }

    B^c = { 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, ...}

    A^c U B^c = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, ... }

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