Recent content by mcampbell

Book XII (12) of Euclid's elements has about 6 propositions which showcase the method of exhaustion. Prop. 2 is an easy one to digest and you can find lots of writing on it as most histories (eves, boyer, edwards) cover this as a representative example of the use of the method of exhaustion in...

1+1=2 might be argued by many to be a report of an observation, but a statement like 123456789012345678+1=123456789012345679 could not be an observation made by a human (perhaps a counting machine?)

are you calling for historians of mathematics to only utter empirically verifiable statements? :)

I am doing a reading next semester and am trying to find a good text to work off of. I am wondering if anyone else has had a semester of Galois/field theory and what text you would suggest.

Also, since it is an operator on R^3 it's injective and onto iff the null space is trivial for a linear operator A the following are equivalent: A is invertible, A is injective, A is onto since it is an operator you need only consider if the null space is trivial, since if it is trivial it is...

I see... In your original post you have b>c. Your deduction is correct.

I'm confused by the first line. Is it A\rightarrowB and B\rightarrowC ? Anyways, you're close to a solution in 8. Do you know a relation between the implication and or operators? In other words, do you know a statement using implication that is logically equivalent to the statement -a v -c ...

If one were to show quaternions could be represented as a matrix and the product defined as standard matrix multiplication, would associativity follow as a consequence of the fact that a matrix represents a linear transformation and the matrix product is functional compostion?

a) you need to check the definition of subspace. Take a look at an element in R^2: (a,b) and an element in R^3: (x,y,z). R^2 is the set of all 2-tuples with real entries and R^3 is the set of all 3-tuples with real entires. I would say R^2 is not a subspace, but I'll leave it to you to justify...

the conditions remind me of the properties of an ideal.

Parts B and C: some hints: How would you interpret the derivative of a function at a point geometrically? Notice also that in part a you have shown dy/dx to be a function of x and y. How can you relate this function to the geometrical interpretation of the derivative?

The fact that a function is undefined at a point does not imply the limit does not exist at that point.

Do you have anymore information about D? What do you know about equations of a plane?

"Does ending with the e=e statement or x'=x' statement show the original statement holds?" No, you've only shown that e=e and x'=x'...profound statements but not what you set out to show. Here's a hint: x' is in G so x' also has a left inverse...what is it? As for the closure, I am...