In the absence of foundation and replacement, we can say that $x$ is an ordinal iff $x$ is hereditarily well-founded (with respect to the relation $\in$) and hereditarily transitive, where $$ \text{$a$ is transitive} \longleftrightarrow \forall x \forall y (x \in y \in a \rightarrow x \in a) $$ ...

Problem 1: $\S$ $=$ $\{$ $(a,\infty)$ $:$ $a\in \mathbb{R}$ $\}$ $\cup$ $\{$ $(-a,\infty)$ $:$ $a\in \mathbb{R}$ $\}$ is a subbasis for $\mathbb{R}$ with the standard topology. Attempt: Clearly, $\S$ covers $\mathbb{R}$. I must show that $\tau$ (the topology generated by a subbasis) is the stan...

How to solve this inequality $e^{(x-2)}>\frac{1}{x-2}$ ? I tried to plot graph and able to get some bound as answer but not satisfy with the approach ? Anyhow we can convert this problem by substitution that $e^{x}>\frac{1}{x}$ Some help needed?

The derivative $f'(x)$ is correlated with $f(x)$ in a certain sense. In fact, $f'(x)$ is a function of $f$, so we could even say that there's a cause-effect relationship. The derivative at a specific point $c$ of the domain, i.e. $f'(c)$, can either be negative or positive. If $f'(c) > 0$, then ...

This answer suggests that laypeople's intuitive notion of the meaning of these words is consistent with the following claims: A cube has 6 faces, 12 edges, 8 vertices. A cylinder has 3 faces, 2 edges, 0 vertices. A cone has 2 faces, 1 edge, 1 vertex. A sphere has 1 face, 0 edges, 0 vertices. ...