1. ## Comparing sets

Hey everyone,

Thanks for taking the time to look at my thread.

I have this question, and a particular part of it is confusing me, I was hoping someone could help me reach the right place.

The question is:

Let A = {(x,y) is element of ZxZ : | x-y | <= 1},
B = {0,1,2,3,4,5}
C = {3,4,5}

Determine |A Intersection (B x C) |

Now I know what im supposed to do. Determine the cardinality of the resultant set of A Intersection (B x C). B and C are right there, however
acquiring the members of A is confusing me. It appears to me that the set in A would have elements consisting of ordered pairs, because of (x,y), but this doesn't seem right to me. Z x Z means the domain of Integers by cartesian product right? so like... (0,0), (0,1), (0,2)... then (1,0), (1,1)...
and so on? I believe the set A is supposed to have singular elements in it, as opposed to pairs, but how do I get this set. I would appreciate any help =] and I apologise for the vagueness of the question. I haven't done maths for a few years and this is my first experience with this branch.

Cheers =]

2. Originally Posted by yoonsi
Hey everyone,

Thanks for taking the time to look at my thread.

I have this question, and a particular part of it is confusing me, I was hoping someone could help me reach the right place.

The question is:

Let A = {(x,y) is element of ZxZ : | x-y | <= 1},
B = {0,1,2,3,4,5}
C = {3,4,5}

Determine |A Intersection (B x C) |

Now I know what im supposed to do. Determine the cardinality of the resultant set of A Intersection (B x C). B and C are right there, however
acquiring the members of A is confusing me. It appears to me that the set in A would have elements consisting of ordered pairs, because of (x,y), but this doesn't seem right to me. Z x Z means the domain of Integers by cartesian product right? so like... (0,0), (0,1), (0,2)... then (1,0), (1,1)...
and so on? I believe the set A is supposed to have singular elements in it, as opposed to pairs, but how do I get this set. I would appreciate any help =] and I apologise for the vagueness of the question. I haven't done maths for a few years and this is my first experience with this branch.

Cheers =]
No, actually, the set A ist clearly a set of pairs. To elaborate a bit: The notation $A = \{(x,y) \in \mathbb{Z}\times \mathbb{Z} : | x-y | \leq 1\}$ means that A is the set of all pairs $(x,y)$ from $\mathbb{Z}\times \mathbb{Z}$ that satisfy the additional condition that $|x-y|\leq 1$, i.e. that their coordinates differ by at most $\pm 1$. Thus the pairs $(0,0), (1,0), (0,1), (-1,0),(7,8), (-8,-7)$ are all in $A$, to give a few examples.
I think the easiest way to determine the cardinality of the the intersection of $A$ with $B\times C$ is to check all the pairs in $B\times C$, whether they are in $\mathbb{Z}\times\mathbb{Z}$ (which is always true) and satisfy $|x-y|\leq 1$ (which may or may not be true): those pairs belong to A as well and are therefore elements of the intersection.

3. $B\times{C}=$ $\{(0,3),(0,4),(0,5),(1,3),(1,4),
(1,5),(2,3),$
$(2,4),(2,5),(3,3),(3,4),
(3,5),(4,3),(4,4),
(4,5),(5,3),(5,4),(5,5)\}$

Let, $A = \{(x,y)\in{Z\times{Z}}$ $:$ $| x-y |\leq{ 1}\}\$

So, $A\cap{(B\times{C})}=\{(2,3),(3,3),(3,4),(4,3),(4,4 ),(4,5),(5,4),(5,5)\}$

4. ## Thanks!

Thank you both very much, I appreciate that you took the time. Looking at it now, it seems very obvious now, and I feel kind of silly.

5. Originally Posted by yoonsi
Hey everyone,

Thanks for taking the time to look at my thread.

I have this question, and a particular part of it is confusing me, I was hoping someone could help me reach the right place.

The question is:

Let A = {(x,y) is element of ZxZ : | x-y | <= 1},
B = {0,1,2,3,4,5}
C = {3,4,5}

Determine |A Intersection (B x C) |

Now I know what im supposed to do. Determine the cardinality of the resultant set of A Intersection (B x C). B and C are right there, however
acquiring the members of A is confusing me. It appears to me that the set in A would have elements consisting of ordered pairs, because of (x,y), but this doesn't seem right to me. Z x Z means the domain of Integers by cartesian product right? so like... (0,0), (0,1), (0,2)... then (1,0), (1,1)...
and so on? I believe the set A is supposed to have singular elements in it, as opposed to pairs, but how do I get this set. I would appreciate any help =] and I apologise for the vagueness of the question. I haven't done maths for a few years and this is my first experience with this branch.

Cheers =]
Hey yoonsi...

A is a set of ordered pairs from the integers just as you supposed. The caveat is that the difference of the two numbers must be <=1 which makes life a little easier.

Since we are going to look at the intersection of A with BxC, and both B and C have integers > 0 and < 5 let's restrict the pairs in A to between 0 and 5.

Thus we have ordered pairs from {0,1,2,3,4,5} which have a difference of <=1.
{(0,0),(0,1),
(1,0),(1,1),(1,2),
(2,1),(2,2),(2,3),
(3,2),(3,3),(3,4),
(4,3),(4,4),(4,5),
(5,4),(5,5)}

Now you can determine BxC to get another set of ordered pairs, then find the intersection.

Hope that helps.