Thank you everyone for being a source of help in previous problems I've posted here. I'm starting an intermediate course in Differential Equations and I'm enjoying it so far, but this one problem on my homework seems to be giving me a problem and I think that I haven't fully grasped the...
Say you have two die D_1 and D_2. If one of them comes up as 1 and the other comes up as n then min{D_1,D_2} is always going to be 1. This is because the minimum of 1 and any other number has to be 1 because you can't roll lower than 1!
Yes I do have some knowledge of direct products. Could I construct a group as such: \mathbb{Z}_{5}\times\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}?
[b]1. Homework Statement [/]
Is there a group G with order 100 such that it has no element of order 4? How would one go about proving the existence of such a group?
[b]2. Homework Equations [/]
For every prime divisor p of a group, there exists an element with order p.
The...
I think I see...?
|ab|\leq\left|\frac{1}{2}(a^{2}+b^{2})-\frac{1}{2}(a+b)^{2}+\frac{1}{2}(a+b)^{2}\right|
\left|ab\right|\leq\left|\frac{1}{2}(a^{2}+b^{2})\right|
\left|ab\right|\leq\frac{1}{2}(a^{2}+b^{2})
Right I've noticed that the values on both sides of the minus sign are all positive, however that doesn't necessarily mean that \left|ab\right| is less than \frac{1}{2}(a^{2}+b^{2}). Because its an absolute value, the LHS of the minus sign could be smaller than the RHS while preserving the...
Homework Statement
Show that \forall a,b \in R:
\left|ab\right|\leq\frac{1}{2}(a^{2}+b^{2})
Homework Equations
Triangle Inequality seems to be useless.
The Attempt at a Solution
(a+b)^{2}=a^{2}+b^{2}+2ab
2ab=(a+b)^{2}-(a^{2}+b^{2})...
Homework Statement
Does a general formula exist?
\sum \limits_{k=0}^{m_1} a_kx^k\cdot\sum \limits_{k=0}^{m_2} b_kx^k=\sum \limits_{k=0}^{m_1+m_2} c_kx^k
Homework Equations
The Attempt at a Solution
I am having trouble understanding the relation between the c coefficients in the product and...
I believe I have found an answer. \forall{X}\in{\tau}:X=X^{\circ}
Where X^{\circ} is the interior of X, and \tau is a topological space. Doing some reading I found a definition of a topology which defines it as a collection of open sets, and then a definition which said that a set is open iff it...
Homework Statement
Given that A and B are in a topology, show that if A is contained in B, then the interior of A is contained in B.
Homework Equations
The interior of A:={a: there exists a neighborhood which is a subset of A}
The Attempt at a Solution
I can prove that the interior of A is...