Hi,
For a set of real numbers, or integers and few more, we can perform basic arithmetic operations like x+y, x-y, xy, x/y.
what's example of a set/s where we can't perform these basic operations?
Thank you.
When we consider sets with operations on them we usually require certain properties. Given a set G with an operation * on it
1) For all a and b in G then c = a*b is also in G. (Closure)
2) Given a, b, and c in G then a*(b*c) = (a*b)*c (Associativity)
3) There is a unique element e of G such that a*e = e*a = a (for every a in G) (Identity)
4) For each a in G there is a unique element [imath]a^{-1}[/imath] such that [imath]a*a^{-1} = a^{-1}*a = e[/imath] (Inverses)
(When * is actually more like an addition we use the symbol + for the group operation. Some groups have both a + and * operation defined on them. With a few more conditions we call this structure a field.)
If these conditions are met then the set G and operation * combine to form an object called a "group." Most objects of Mathematical interest are groups. We can add "structure" to this as well... Commutative groups have the additional property a*b = b*a.
So, to address your question, it doesn't really make a lot of sense to define an operation such that your set would not be closed. If we can define a*b then we usually just add the element c = a*b to the set.
Now, there are some sets where division is not impossible, but difficult to work with. These are called "modular groups." (These are sometimes called "remainder groups.") For example, consider an old fashioned clock with hands and everything. It reads time from Midnight = 0 to Noon = 12, and then resets to 0. So if we add the two times, 3 + 4 we get 7, and if we add the two times, 7 + 8 we get 7 + 8 = 15 which resets to 3 (because 7 AM plus 8 hours is 3 PM.) We can multiply here, too. 3* 3 = 9 and 5*4 = 20 = 8. We can define division but because fractions are forbidden (they don't belong to the set) we run into some problems with some numbers. The multiplicative inverse of 5 is 5 because 5*5 = 25 = 1 (recall that an inverse is an element [imath]a^{-1}[/imath] such that [imath]a*a^{-1} = e[/imath].) So, for example, 7/5 would be defined to be [imath]7/5 =7*5^{-1} = 7*5 = 35 = 11[/imath]. (You can check this easily... if 7/5 = 11, then 11*5 = 55, but 55 resets to 7 so 11*5 = 7 as expected.) But how can we divide by, say, 2? There is no element in our set such that 2*a = 1. Yes, we can divide 10/2 and get 5. But we cannot legally do the process this way, we have to be able to do it for more than even numbers. How would we do 7/2, for example? 2, 3, 4, 6, 8, and 9 have no multiplicative inverses in this system so division is pretty dodgy. "Clock Math" is the group [imath]\mathbb{Z}_{12}[/imath]. Other modular groups work the same way and can be very useful.
-Dan