False. A set may contain just some of its boundary points, making it neither open nor closed.
2.
Any set which is symmetric in the origin is invariant under a mirror reflection.
True ; False
False. A capital S is symmetric about its centre, but has no mirror symmetry. Note that a half turn is the product of two reflections in perpendicular axes.
3.
The final line of the Non-Overlap Theorem can be written: If A(R) 1 then no two regions R overlap.
True ; False
False. As an extreme case, it is easy to find two lattice centred line segments (with area 0) which overlap.
4.
Describe a set R for which P'(–x, –y) is not a point of R for all P(x, y) in R.
For example, an open semicircular region on (–1, 0), (0, 1) as diameter. Or the circular disk of radius 1 centred at (2, 2).
5.
The contrapositive of a theorem is the same as the converse.
True ; False
False. For example: Theorem: If S is a square then S has four equal sides. Converse: If S has four equal sides then S is a square (false). Contrapositive: If S does not have four equal sides, then S is not a square (true).
6.
Check out the proof of Minkowski’s Theorem. Do we need both the convex and symmetry properties?
Yes ; No
Yes, both properties are used in the proof. (Of course, there is the possibility that a different proof might not need these. But simple examples show that then the theorem fails.)
1 then no two regions R overlap.