MathDB

Problem 4

Part of 2001 VJIMC

Problems(2)

A+B subset A+C, A bounded, C closed, convex

Source: VJIMC 2001 1.4

7/15/2021
Let A,B,CA,B,C be nonempty sets in Rn\mathbb R^n. Suppose that AA is bounded, CC is closed and convex, and A+B⊆A+CA+B\subseteq A+C. Prove that B⊆CB\subseteq C. Recall that E+F={e+f:e∈E,f∈F}E+F=\{e+f:e\in E,f\in F\} and D⊆RnD\subseteq\mathbb R^n is convex iff tx+(1−t)y∈Dtx+(1-t)y\in D for all x,y∈Dx,y\in D and any t∈[0,1]t\in[0,1].
set theory
ring arithmetic, x^n=x for all x implies x,y^(n-1) commute

Source: VJIMC 2001 2.4

7/21/2021
Let RR be an associative non-commutative ring and let n>2n>2 be a fixed natural number. Assume that xn=xx^n=x for all x∈Rx\in R. Prove that xyn−1=yn−1xxy^{n-1}=y^{n-1}x holds for all x,y∈Rx,y\in R.
Ring Theoryabstract algebra