Find a subset of R^2 which is not of the form AxB where A and B are subsets of R.
To find a subset of ℝ² that is not of the form A × B, where A and B are subsets of ℝ, we can consider a simple example.
Let's define the set S as S = {(x, y) ∈ ℝ² | x = y}.
Here, S is the subset of ℝ² consisting of all points where the x-coordinate is equal to the y-coordinate.
To show that S is not of the form A × B, we need to prove that for any subsets A and B of ℝ, S cannot be expressed as the Cartesian product A × B.
By definition, the Cartesian product A × B consists of all ordered pairs (a, b), where a is an element of A and b is an element of B. However, for S, the elements are only in the form (x, y) with x = y.
If we assume that S can be expressed as A × B, then it implies that there exist elements a and b such that (a, b) ∈ A × B, where a ≠ b. This contradicts the definition of S, which requires the x-coordinate to be equal to the y-coordinate.
Therefore, we can conclude that S is a subset of ℝ² that is not of the form A × B, where A and B are subsets of ℝ.