Is it true that $\sup(Y) - \sup(X) \leq \sup(Y \setminus X)$

Question

Let $X$ and $Y$ be two subsets not empty of $\mathbb{R}$ such that $X \subset Y$

Is it true that: $$ \sup(Y) - \sup(X) \leq \sup(Y \setminus X) $$

thanks.

Answer 1

Of course this is false. Look at an example - if that example is not a counterexample you got very unlucky, look at another example.

There's really no reason I can imagine why anyone would think this was true; hence the question: Where did this question come from? I actually formulated a hypothesis about that. It is true that$$\sup(Y)-\sup(X)\le\sup(Y-X)$$(at least if $X\ne\emptyset$), where $$Y-X=\{y-x:y\in Y, x\in X\}.$$And unfortunately people do sometimes write $Y-X$ when they should write $Y\setminus X$. So I suppose someone could see that true formula and assume that $Y-X$ meant $Y\setminus X$...