# If we have two sets A = { x belongs to Z, with property 3/(x-2) belongs to ZB = {x belongs to Z , with property ( x-2)/3 belongs to Z . Find A, B, A-B .

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x is an element of A where x belongs to z

That means that x should be an integer (1,2,3...)

We have 3/(x-2) is an integer , then (x-3) should be divided by 3.

Then (x-2) should be either 3,-3, 1, or -1

x-2=3 ==> x=5

x-3=-3 ==> x=0

x-2=1 ==> x=3

x-2=-1==> x=1

Then A elements are 0,1,3, 5

A= {0,1,3,5}

Now for B. B elements should verify (x-2)/3

Since x belongs top z, then x-2)/3 should be an integer, that means (x-2) is a multiplex of 3

(x-2)= 3n where n=...-3,-2,-2,1,0,1,2,3....

x-2= 3(-2) ==> X=-4

X-2=3(-1) ==> X=-1

x-2= 3(0) ==> x=2

x-2=3(1) ==> x= 5

x-2=3(2) ==> x=8

x-2=3(3) ==> x= 11

==> B = {...-4,-1,2,5,8,11,...}

Then A-B elements are all elements in A without elements in B:

A-B = A - (A intersect B)

A-B = {0,1,3,5}-{5} = {0,1,3}

By the criteria mentioned 3/(x-2) = n where n is a ehole number

So (x-2)/3 = 1/n

Or

x = 3/n +2.

Therefore , A = { (3/n)+2| n is an integer.

(x-2)/3 is a whole number. Or (x-2)/3 = n Or x = 3n+2.

So A = { (3/n)+2} for all integers , but n is not zero

B = {3n+2} .

The elenments of B in A are when n =1 and when n =-1. So,

A-B = {3/n+2} - {3} - {-1} .

In order to find A elements, we have to follow the given property of the set.

The ratio 3/(x-2) belongs to Z, if and only if the result of division is a integer number. For that, the denominator must be a divisor of 3, which means (x-2)=1, or (x-2)=-1, or (x-2)=3, or (x-2)=-3

x-2=1

x=2 +1=3

x-2=-1

x=2-1=1

x-2=3

x=2+3=5

x-2=-3

x=2-3

x=-1

A={-1,1,3,5}

In order to find B elements, we have to follow the property of the set.

(x-2) has to be a multiplex of 3, so that the ratio to be an integer number.

x-2 = 3k

x=2+3k

Now, we'll plug in values for k, in order to find out x values.

k=-3

x=2 + 3*(-3)=2 -9=-7

k=-2

x= 2 + 3*(-2)= 2 -6=-4

k=-1

x=2 + 3*(-1)=2-3=-1

k=0

x=2 + 3*(0)=2

k=1

x=2 + 3*(1)=2+3=5

B={.....-7,-4,-1,2,5,....,2+3k}

The difference between the 2 sets, A-B=what elements are in A set and are not in B set= {1,3}.