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### Notation for the derivative of a function: $f'$ or $f'(x)\;$?

The derivative of a function is often defined as $f'$ and $f'(x)$. So which one is it? $f'(x)$ is the output of the function $f'$, so why do I see people using $f'$ and $f'(x)$ interchangeably to ...
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The axiom of regularity says: (R) $\forall x[x\not=\emptyset\to\exists y(y\in x\land x\cap y=\emptyset)]$. From (R) it follows that there is no infinite membership chain (imc). Consider this set: $... 2answers 1k views ### Which rule is applied to define the operator precedence for factorial Please apologize the question, I struggled with finding a good formulation in the first place: Looking at$\binom{2n}{k}$it is very clear that for n,k integer and n>k we can solve it by calculating: ... 2answers 278 views ### Proof verification: prove$A\subseteq B$if and only if$A\cap B=A$. Can someone verify whether my proof is logically correct? :) Proof: Assume$A\subseteq B$. Then for every element that belongs in A, such element also belongs in B. Then$A\cap B \subset A$. If$x \...
The question says : If $f:X\rightarrow Y$ and $a,b \subseteq X$, then $f(a \cap b)$ is equal to? - $f(a)-f(b)$ - $f(a)\cap f(b)$ - a proper subset of $f(a)\cap f(b)$ - $f(b)-f(a)$ I'm ...