In a set we have $a(b+c)=ab+c$. What is it?

Question

suppose $A\subseteq \mathbb{N}$ and for any $a,b,c\in A$ with $a<b<c$ we have $$a(b+c)=ab+c$$ what are all $A$ with this property?!

here $\mathbb{N}=\{1,2,3,...\}$.

We have $ac = c$, so with $a < b < c$ this forces $A$ to have at most 3 elements and $1 \in A$ (or it can have less that three elements, and then the premise is false). — dtldarek, Feb 11 '13 at 22:43

Ross Millikan · Accepted Answer · 2013-02-11T22:48:28.037

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Apply the distributive property, subtract the common term, and you get $ac=c$, so $a=1$. Any three element subset that includes $1$ will force this. Any set with less than three elements will also meet the requirement as there is no $a,b,c$.

edited Feb 11 '13 at 22:48

answered Feb 11 '13 at 22:42

Ross Millikan

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2

What about $A = \varnothing$? – dtldarek Feb 11 '13 at 22:46

In a set we have $a(b+c)=ab+c$. What is it?

1 Answers1