Geometric Puzzle: Find P & Q in R2 Square

In summary, the sets P and Q that satisfy the given conditions are P = UR ∪ LL ∪ UA and Q = LR ∪ UL ∪ LA, where each set is defined as a subset of X = [-1,1] x [-1,1]. These sets are completely contained in the square with vertices (1,1), (1,-1), (-1,-1), and (-1,1), and they are disjoint. Furthermore, they are both connected sets, satisfying all four given properties. These sets cannot be path-connected, as any continuous path from (1,1) to (-1,-1) would have to intersect a path from (-1,1) to (1,-1).
  • #1
HallsofIvy
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Find two sets P and Q satisfying
I) P and Q are completely contained in the square, in R2, with vertices (1, 1), (1, -1), (-1, -1), and (-1, 1).
II) P contains the ponts (1, 1) and (-1, -1) while Q contains (-1, 1) and (1, -1).
III) P and Q are disjoint.
IV) P and Q are both connected sets.
 
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  • #2
HallsofIvy said:
Find two sets P and Q satisfying
I) P and Q are completely contained in the square, in R2, with vertices (1, 1), (1, -1), (-1, -1), and (-1, 1).
II) P contains the ponts (1, 1) and (-1, -1) while Q contains (-1, 1) and (1, -1).
III) P and Q are disjoint.
IV) P and Q are both connected sets.
P and Q cannot be path-connected, because any continuous path from (1, 1) to (-1, -1) would have to intersect one from (-1, 1) to (1, -1). So the problem has to be about the topological definition of connectedness rather than the geometric notion of path-connectedness, and we must look for sets that are connected but not path-connected.
[sp]Let $X = [-1,1]\times[-1,1]$, and define subsets of $X$ by $$UR = \{(x,y)\in X : x>0,\ y > \tfrac12\sin\tfrac1x\},$$ $$LR = \{(x,y)\in X : x>0,\ y < \tfrac12\sin\tfrac1x\},$$ $$UL = \{(x,y)\in X : x<0,\ y > \tfrac12\sin\tfrac1x\},$$ $$LL = \{(x,y)\in X : x<0,\ y < \tfrac12\sin\tfrac1x\},$$ $$UA = \{(x,y)\in X : x=0,\ y > 0\},$$ $$LA = \{(x,y)\in X : x=0,\ y < 0\},$$ (the names of the sets are meant to indicate Upper Right, Lower Left, etc., and A denotes $y$-Axis). These sets are all disjoint, and each of them is connected.

Let $P = UR\cup LL\cup UA$, $Q = LR\cup UL\cup LA.$ Then properties I), II), III) certainly hold. To see that IV) also holds, suppose that $U$ and $V$ are disjoint open sets with $P\subset U\cup V$. Since the three component parts of $P$ are connected, each of them must lie entirely within one of the sets $U$, $V$. In particular, $UA\subset U$ say. Then $U$, being open, must contain a neighbourhood extending each side of the positive $y$-axis and therefore contains points in both $UR$ and $LL$. But those sets are both connected, and it follows that $U$ must contain the whole of $P$, so that $V$ is disjoint from $P$. That shows that $P$ is connected; and a similar argument shows that so also is $Q$.[/sp]
 
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FAQ: Geometric Puzzle: Find P & Q in R2 Square

1. What is a geometric puzzle?

A geometric puzzle is a problem or challenge that involves using geometric concepts and principles to find a solution. These puzzles often require critical thinking and problem-solving skills.

2. How do you solve a geometric puzzle?

The first step in solving a geometric puzzle is to carefully read and understand the given problem. Then, use your knowledge of geometric principles and techniques to try and find a solution. You may also need to use trial and error and try different approaches until you find the correct solution.

3. What is R2 Square?

R2 Square is a term used in mathematics to refer to a two-dimensional space, also known as a plane. The "R" stands for real numbers, meaning that the space includes all possible points with both x and y coordinates that are real numbers.

4. How do you find P and Q in R2 Square?

In the context of a geometric puzzle, finding P and Q in R2 Square means finding the coordinates of two points in a two-dimensional space. This can be done by using geometric principles such as distance formulas, slope calculations, or by graphing the given points and visually determining their coordinates.

5. Why are geometric puzzles important?

Geometric puzzles are important because they help develop critical thinking, problem-solving, and spatial reasoning skills. They also allow us to apply and strengthen our understanding of geometric concepts and principles in a fun and engaging way.

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