wcwcwc asked in 科學數學 · 1 decade ago

logic and counting questions

1). let U={1,2,3,...,2008}

how many integers in U can be divided (evenly) by at least on of the 2,3,5 ???

2). for sets A,B,C,D (are subset of) U, prove of disprove the following

a). if A (is a subset of) B and B(is not a subset of) C, then A(is not a subset of) C

b). A-B = compliment of (B-A)

c). if A and B are disjoint and C and D are disjoint, then A and C, B and D are disjoint

d). A (is a subset of) B if and only if A and complement of B = empty set

e). A (is a subset of) B if and only if complement of A or B = U

f). P(A or B) = P(A) or P(B)

*P() = powerset

2 Answers

Rating
  • 蜉蝣
    Lv 6
    1 decade ago
    Favorite Answer

    1). 2008/2=1004,2008/3=669...,2008/5=401..

    2008/6=334...,2008/10=200..,2008/15=133...,2008/30=66...

    The total =(1004+669+401)-(334+200+133)+66=1473

    a). if A (is a subset of) B and B(is not a subset of) C, then A(is not a subset of) C

    False. Consider A={1},B={1,4},C={1,2,3},then

    A (is a subset of) B and B(is not a subset of) C, but A(is a subset of) C

    b). A-B = compliment of (B-A)

    False. Consider U={1,2,3,} ,A={1,2},B={1,3},then

    A-B={2},B-A={3},compliment of (B-A)=U-(B-A)=U-{3}={1,2}

    {2}is not equal to {1,2}

    c). if A and B are disjoint and C and D are disjoint, then A and C, B and D are disjoint

    False. Consider A={1,2},B={3,4},C={1,3},D={2,4}.Then it's clearly that

    A and B are disjoint and C and D are disjoint, but A and C, B and D are not disjoint

    d). A (is a subset of) B if and only if A and complement of B = empty set

    True. (Actually,it's clearly true by using set-graph.)

    =>)Let x be an element in A--#. Since A (is a subset of) B,we have x in B.

    Then x doesn't belong to complement of B--##.

    Hence A and complement of B = empty set (by#and##)

    <=)Let x be an element in A--#. SinceA and complement of B = empty set,

    we have x doesn't belong to complement of B .

    In the other words,x belongs to B##

    Hence A (is a subset of) B (by#and##)

    e). A (is a subset of) B if and only if complement of A or B = U

    True.

    =>)If x is an element in A,since A (is a subset of) B,we have x in B--##.

    If x is not an element in A,then x belong to complement of A---##.

    Hence we have complement of A or B = U (by#and##)

    <=)Let x be an element in A--#. Since complement of A or B = U ,and x doesn't belong to complement of A

    we have x belongs to B-## .

    Hence A (is a subset of) B (by#and##)

    f). P(A or B) = P(A) or P(B)

    power set ,I see

    False. Consier A={1},B={2}.

    Then P(A or B)=P({1,2})={{},{1},{2},{1,2}}

    P(A)={{},{1}},P(B)={{},{2}},P(A) or P(B)={{},{1},{2}}

    But,P(A or B) is not equal to[ P(A) or P(B)]

    2008-10-13 18:42:22 補充:

    You're totally right. Well-done.

  • 1 decade ago

    thx,

    i checked your method, and i draw a venn diagram and i understand now.

    but just for the once last thing,

    if i say let

    A be the number can be divided by 2,

    B .... 3,

    C ... 5,

    A and B be the number can be divided by 2X3 = 6

    A and C = 10

    B and C = 15

    A and B and C is 30

    2008-10-13 02:53:52 補充:

    so my finally answer would be looking for

    (A or B or C)=1473

    is that right?

    or my "and" ,"or" in wrong place?

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