# Cardinality of these sets!

• Aug 4th 2010, 06:12 AM
Aeonitis
Cardinality of these sets!
Hey guys, this is my first post, (Hi) was just wondering if i could get your help. I'm studying for my repeats and you guys can save me.

If X = {1,2,3,4}, Y = {2,4,6} what is the cardinality of the following sets?

(i) A = {x|x mod 2 = 0 and 0 <=x<=20}
(ii) B = X * X * Y
(iii) C = {(x,y)|x ≠ y and x,y ∈ X}

Please explain your train of thought in solving this. I am trying hard to understand the right way to approach this question quickly, Thank you for your time guys...
• Aug 4th 2010, 06:49 AM
Aeonitis
I've tried them out, with the following answers:-

(i) A = {2,4,6,8,10,12,14,16,18,20}, Therefore A has a cardinality of 10 (elements).
(ii) B = Cartesian Product of 'X times X times Y' or better yet '4 by 4 by 3' elements each to give a total of 48 in cardinality?!
(iii) C = UNSOLVED!!!

I wanna make sure someone agrees with me having the right answers since you're the pros

I really wanna know what the '|' symbol stands for or means, as in 'x|x'. Hard to specifically search for in a book.
• Aug 4th 2010, 07:16 AM
Plato
Quote:

Originally Posted by Aeonitis
I've tried them out, with the following answers:-
(i) A = {2,4,6,8,10,12,14,16,18,20}, Therefore A has a cardinality of 10 (elements).
(ii) B = Cartesian Product of 'X times X times Y' or better yet '4 by 4 by 3' elements each to give a total of 48 in cardinality?!
(iii) C = UNSOLVED!!!
I really wanna know what the '|' symbol stands for or means, as in 'x|x'. Hard to specifically search for in a book.

Parts a & b are correct.
For part c: $|X\times X|-|X|=16-4=12$.

For integers $x|y$ usually means x divides y.
But it can mean other things.
What is the context of you question?
• Aug 4th 2010, 08:04 AM
Aeonitis
Quote:

Originally Posted by Plato
For part c: $|X\times X|-|X|=16-4=12$.

For integers $x|y$ usually means x divides y.
But it can mean other things.
What is the context of your question?

I guess there's a chance that De Morgan's Law might plays a part in this since before this question, i was asked to write down De Morgan's Laws for sets. Found a way to type the symbols accurately, To put it precisely, the 'C' question is

C = {(x,y)|x ≠ y and x,y ∈ X}

I do know that '/' means divide, but i'm assuming that the symbol '|' plays a different part, am I wrong?
• Aug 4th 2010, 08:10 AM
Plato
Quote:

Originally Posted by Aeonitis
C = {(x,y)|x ≠ y and x,y ∈ X}
I do know that '/' means divide, but i'm assuming that the symbol '|' plays a different part, am I wrong?

In that context the “|” is read as “such that”: The set of ordered pairs (x,y) such that x is not y and…”
• Aug 4th 2010, 08:13 AM
HallsofIvy
"x/y" means the number "x divided by y" but it is more common to use "|" for the statement "x|y" meaning "x divides evenly into y (has remainder 0)".

Here, however, it is clear (to me, anyway) that the "|" is just separating the pairs (x,y) from the condition " $x\ne y$". This set is the set of all pairs (x, y) from X (so x and y can be any of 1, 2, 3, 4) such that x is not equal to y.

X itself has 4 members so $X\times X$ has 16 members. Requiring that $x\ne y$ drops 4 of those: (1, 1), (2, 2), (3, 3), and (4, 4). X contains 16- 4= 12 members.
• Aug 4th 2010, 08:27 AM
Aeonitis
Thank you guys, i will post the full answer for future questioneers ;)

(i) A = {2,4,6,8,10,12,14,16,18,20}, Therefore A has a cardinality of 10 (elements).
(ii) B = Cartesian Product of 'X times X times Y' or better yet '4 by 4 by 3' elements each to give a total of 48 in cardinality?!

(iii) C = {(x,y)|x ≠ y and x,y ∈ X}

pairs x,y {such as (1,1),(1,2),etc...} drawn from set X with a cardinality of '4 by 4 = 16' as in the question "x,y ∈ X". Due to the statement 'x ≠ y' pairs can't come in equals, discarding the following four sets (1,1),(2,2),(3,3),(4,4). The end product is 16-4 giving a cardinality of 12 for set 'C'.

Thank you so much Guys :D