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The assignment reads sketch the following subsets of $\mathbb{R}^2$:

(a) $\{(x, y) \in \mathbb{R}^2 \mid x > [y]\}$

(b) $\{(x, y) \in \mathbb{R}^2 \mid |x|^p + |y|^p < 1\}~\mathrm{for}~p = \frac{1}{2}, 1, 2, 4$

(c) $\{(x, y) \in \mathbb{R}^2 \mid \max\{|x − 1| , |2y − 6|\} < 4\}$

These are from a question sheet from a online course I'm doing. I have this from the notes.

For a real number $x$ we write $[x] := \max\{k \in \mathbb{Z}, k \leq x\}~\mathrm{and}~\{x\} = x − [x]$ for the integer part (or floor) and the fractional part of $x$. Also recall the definition of the absolute value. For a real number $x$, $|x| := \{x~\mathrm{if}~x \geq 0~\mathrm{and}~−x~\mathrm{if}~x \leq 0\}$.

But that's about it for the question. I think I have part (b) figured out I just seem to be very confused about part (a) and (c) if anyone could maybe explain that would be so helpful.

Mike Pierce
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phil demeanour
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1 Answers1

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For problems of this type the following strategies should prove useful:

  1. Replace inequalities with equalities to get a description of the boundary of the set you are looking for.

  2. An inequality involving a maximum can be split into separate inequalities for each element of the maximized set. (For example: $\max\{a,b\}<c \iff a<c$ and $b<c$, but $\max\{a,b\}>c \iff a>c$ or $b>c$.)

  3. The logical operators "and", "or", and "not" all have set-theoretic analogues.

hife
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