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

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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.

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?