Maths L.C.

Patriciamc93

Hi,

Just wondering if anybody is doing the groups option for leaving cert maths higher level! I am on the first section and already lost.
I don't really understand the four properties, especially the first 2:

1.Closure
2.Associativity
3.The Identity Element
4.The Inverse element

IF anybody could help it would be much appreciated,
Thanks!

MathsManiac

A set is closed under a binary operation if, when you do the operation to two elements of the set, you always end up with an element of the set.

For example, the set of all integers (positive and negative whole numbers) is closed under addition: if you add two integers you always end up with an integer. It's not closed under division, because if you divide two integers you don't necessarily end up with an integer (e.g. 3 divided by 2 is not an integer).

A binary operation * is associative if it has the property that (x*y)*z = x*(y*z), for all x, y, and z in the set. For example, addition is associative, because (x+y)+z is always equal to x+(y+z). Subtraction is not associative, because (x-y)-z is not always equal to x-(y-z).

MathsManiac

Perhaps I should also have made clear that you can't consider the issue of closure without thinking about both the operation and the set.

For example, even though the set of integers is not closed under division, the set of non-zero rational numbers is closed under division. (A rational number is any number that can be written as a fraction whose numerator and denominator are integers, and the denominator is not zero.) When you divide one non-zero fraction by another, you always end up with a non-zero fraction.