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January 9[edit]

How to prove the commutative property?[edit]

Also, how to interpret a natural number, an integer, a real number, a complex number? FrankDev (talk) 01:18, 9 January 2013 (UTC)[reply]

It depends on which commutative property you want to prove, and what you're assuming. Implementation of mathematics in set theory might have something relevant, or might be completely orthogonal to what you're interested in — depending on what you're interested in. --Trovatore (talk) 01:33, 9 January 2013 (UTC)[reply]

See natural number, integer, real number, and complex number, then ask any specific question you have. StuRat (talk) 01:45, 9 January 2013 (UTC)[reply]

One of the caveats of proofs is that you have to take assumptions before you can derive conclusions. Also, as others have said, it depends on the operation you are trying to prove you can commute. Exponentiation is not commutative, for example. i kan reed (talk) 20:38, 9 January 2013 (UTC)[reply]

modulus/cardanality[edit]

Are the modulus of a number and a sets cardinality the same or are they just unrelated and use similar notation? — Preceding unsigned comment added by 109.153.170.141 (talk) 21:05, 9 January 2013 (UTC)[reply]

I wouldn't say unrelated but they are different ideas sharing the same notation. |x| is the absolute value of x; |A|, sometimes written #A is the number of elements in a set A.RDBury (talk) 21:28, 9 January 2013 (UTC)[reply]

The notation |A| is also used to denote the measure of A, in a given measure space, and it is sometimes used for other positive quantities attached to a given class of objects. A common feature of all these different notions is some form of subadditivity: the modulus satisfies |x+y|≤|x|+|y|, while cardinality and more generally measures satisfy |A∪B|≤|A|+|B|. --pm a 22:35, 9 January 2013 (UTC)[reply]

Interesting point. I wonder whether that had anything to do with the choice of notation, which otherwise I would probably have called coincidental. --Trovatore (talk) 22:46, 9 January 2013 (UTC)[reply]

Matrix inversions[edit]

I have a Matrix, M. I invert it to get another, N.

MN=The identity matrix

But what does NM equal? Is it also the identity matrix? — Preceding unsigned comment added by 109.153.170.141 (talk) 22:45, 9 January 2013 (UTC)[reply]

Yes, because the inverse of a nonsingular matrix must be unique and not depend upon whether it is left- or right- multiplied.

To see why this is, let M be a square matrix and suppose there exists N_r and N_l such that N_r is the "right inverse" of M, that is, right-multiplying by N_r gives the identity, and N_l is the left inverse. MN_r=I and N_lM=I, in simpler terms. Now left-multiply both sides of the first expression by N_l, which is the "left-inverse," and get N_l(MN_r)=N_l. But matrix multiplication is associative so N_l(MN_r)=(N_lM)N_r=N_l. We defined N_l to be the left inverse so N_lM=I, which implies that N_l=N_r. 72.128.82.131 (talk) 23:02, 9 January 2013 (UTC)[reply]