Math Is Fun Forum

What is a number?
I mean there must be some kind of meaning to just a number, otherwise things like number algebra could just have been made up?

A matrix is for the most part meaningless by itself, just like a number is.  But we can assign meaning to numbers by have the numbers represent certain things: such as the number of apples I have in my fridge.  It is the same way with matrices.  Matrices can be used to represent certain things in specific contexts: A linear transformation, a graph, or the pixels in a photo.

These proofs typically go through without any difficulty; everything comes straight from definition.  What step are you having trouble on?

For the first one, I assume you're also told that 1 is in S?  If there are natural numbers not in S, remember you can always find the smallest one not in S.

For the 2nd one, start writing up the induction here and stop whenever you don't see what to do.

An alternate proof is to let g be an odd permutation, and note that {g, A_n} generates S_n.  Now we have by the diamond (second) isomorphism theorem:

The left side is 2, giving the right side is 2, and you're done.

Looks good, I suppose an attempt at  proof would be just a little too much?  It is rather surprising how elementary the proof is.

The IVT says a lot more than "you will be at the same height". If you walk in a circle (this works for any parametrized continuous path, but circle is easiest to see), let your height at time t be described by h(t).  Here we are walking around the circle in 1 unit time of time.  Then there will exist at least one point which will be exactly the same height as the opposite side of the circle.

We can define a function:

g(t) = h(t) - h(t+1/2)

If g(0) = 0, then we're done: our starting point is at exactly the same height as the opposite side.  Otherwise, g(0) is either positive or negative.  But g(1/2) has exactly the opposite sign of g(0):

g(0) = h(0) - h(1/2)
g(1/2) = h(1/2) - h(0)

(Remember that h(0) = h(1).  They are exactly the same point!)

Now the IVT says that g(t) must be zero at some point in between 0 and 1/2.  And that's it.

Of course, probably too difficult to put in the page...

I was referring to the length of input that I was giving, and if your next line is your input, it wasn't.

Since B' - A is contained in B', you can easily simplify the inside to just one term.  Try to take it from there.

It seems like you are posting your homework so others can do it for you.  Please show us what you're thinking thus far on the problem.

Have you tried the problem at all?  What are your thoughts?

What happens for -1 < p < 0?

What you want to show is that there are two vertical asymptotes, and the inside has one side going up to infinity and the other side goes down to negative infinity.  You can show that this is the case for p < 0, p ≠ 1 by limits.

The only way I could see this occurring is if your model is not accurate.