On dimension of algebraic sets February 9

Let $k$ be an algebraically closed field and $m\leq n$. Suppose $\pi:\mathbb{A}^n\to \mathbb{A}^m$ is map which sends $(a_1,\ldots,a_n)\to (a_1,\ldots,a_m)$. If $V$ is an affine algebraic set, then is it true that $\dim(V)\geq \dim(\overline{\pi(V)})

Prove this simple graph is not planar. February 9

Prove this simple graph is not planar.
I need to show this graph is not planar. I've attempted to find $K_5$ and $K_{3,3}$ as a subgraphs but haven't been successful yet. It's possible but unlikely this graph is planar but I have ...

Is there no difference in symbols between the floor and the ceiling of x February 9

Is there no difference in symbols between the floor and the ceiling of x
Source: Discrete Mathematics with Applications, Susanna S. EppThe symbol of floor of x is [x] and so is the symbol [x] of ceiling of x. Is it correct that there's no difference in symbols be ...

How to find integral of the form $e^xf(x)$ February 9

I always face trouble with these type of integrals.I need to find $$\int{e^x \frac{x(\cos x -\sin x)-\sin x}{x^2}}dx$$My problem would be solved if can express $f(x)$ like $g(x)+g'(x)$ but identifying $g(x)$ by trial and error method is sometimes ted

Square Free congruence modulo n February 9

I am trying to show that if $a^n\equiv a\pmod n$ for all integers $a$ that $n$ is square free. I have an idea to start with the contradiction that suppose $n=p^2m$ for some prime $p$, then n does not divide $a^{p^2m}-a$ for some integer $a$. Any hint

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