query: Non trivial examples of $f\circ g = g \circ f$ but $f^{-1} \neq g$ and $f\neq\mathrm{id}\neq g$.
positive_passages.docid: 24126
positive_passages.text: When functions commute under composition
negative_passages.docid: 33593
negative_passages.text: An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
negative_passages.docid: 155935
negative_passages.text: Examples where $H\ne \mathrm{Aut}(E/E^H)$
negative_passages.docid: 95299
negative_passages.text: What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15
negative_passages.docid: 157597
negative_passages.text: Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$
negative_passages.docid: 101042
negative_passages.text: Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$.
negative_passages.docid: 146152
negative_passages.text: Characterising functions $f$ that can be written as $f = g \circ g$?
negative_passages.docid: 45885
negative_passages.text: If there exist sequence such that $g(x_n)=f(x_{n+1})$, then we have $g(x_0)=f(x_0)$ for some $x_0$
negative_passages.docid: 6098
negative_passages.text: Let $f:A\rightarrow B$ and $C\subset B,D\subset B$. Prove that $f^{-1}(C\backslash D)=f^{-1}(C)\backslash f^{-1}(D)$
negative_passages.docid: 22758
negative_passages.text: If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.
negative_passages.docid: 63836
negative_passages.text: Show there exists $g:(0,1)\to\mathbb{R}^2$ such that $h\circ g$ constant, $g$ continuously differentiable and $g$ injective
negative_passages.docid: 50655
negative_passages.text: Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$
negative_passages.docid: 112263
negative_passages.text: an example for $\pi$-quasinormal subgroup but is not normal
negative_passages.docid: 41363
negative_passages.text: when is f@g not the same as f[g]?
negative_passages.docid: 110643
negative_passages.text: What are the strategies I can use to prove $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$?
negative_passages.docid: 18003
negative_passages.text: Standard example where Jacobson radical not equal nilradical
negative_passages.docid: 76749
negative_passages.text: Example of a finite non-commutative ring without a unity