Please prove both A and B

A)

Hint: Cover S with small rectangles at the origin and a union of four parameterization

B)

Prove that P(E) = 1 using A and explain why E illustrates this heuristic principle.

Fix B > 0. Let 12 = [-B,B]4 and let & be the set of Jordan measurable subsets of 22. Let P be the uniform probability measure on 12. Prove that the set S = {(a, b,c,d) E22:ad bc = 0} has zero Jordan measure. Define an event E E which illustrates the heuristic principle of "generic" vectors in two dimensions: If you randomly pick 2 vectors in [-B,B]? then they will form a basis for R2 with 100% likelihood. Fix B > 0. Let 12 = [-B,B]4 and let & be the set of Jordan measurable subsets of 22. Let P be the uniform probability measure on 12. Prove that the set S = {(a, b,c,d) E22:ad bc = 0} has zero Jordan measure. Define an event E E which illustrates the heuristic principle of "generic" vectors in two dimensions: If you randomly pick 2 vectors in [-B,B]? then they will form a basis for R2 with 100% likelihood