Lesson 3

Such geometric interpretations often turn out to be useful for establishing bijections. Let us give a few examples. In all these examples, we will establish the equinumerosity of two subsets \(A, B \subseteq \mathbb{R}^{2}\) of the plane. Each time, we will implicitly use the following observation: the cardinality of a set does not change under translation and scaling.

• To show that any two circles (with non-zero radius) are equinumerous, we place one circle inside the other.

  



• We can also establish the equinumerosity of a circle and a triangle.

  



• To show that any two segments (with positive length) are equinumerous, we can place them one below the other.

  



• Finally, to prove that the open interval (for example, \((0,1)\)) is equinumerous to the line, we need to place this interval above the line, breaking it in half.

  

Let us also show that the interval \((0,1)\) is equinumerous to the half-interval \((0,1]\). To do this, we will select a countable subset \(\{1, 1/2, 1/3, 1/4, \dotsc\}\) from both sets and establish a bijection on it with a shift of one (as in the Hilbert hotel). All other points will map to themselves.