For each of the subsets of RxR determine whether it is equal

For each of the subsets of RxR, determine whether it is equal to the cartesian product of two subsets of.

a) {(x,y) | x is not an integer and y is an integer}

b) {(x,y) | x^2 + y^2 < 1 }

c) {(x,y) | y>x}

Solution

Answer :

a) Cosider the set A = {(x,y) | x is not an integer and y is an integer}

Then the set A can be expressed as A = Z^* x Z where Z^* = R - Z

b) Cosider the set B = {(x,y) | x² + y² < 1 }

This cannot be written as a Cartesian product of two subsets of R. For instance the points ( 0.9 , 0.1) and ( 0.1 , 0.9 ) both are inside the set B. Since (0.1)² + (0.9)² =0.82 <1. However, if the given set could be written as a Cartesian product, then

(0.9 , 0.9) should also be in the set. But it is not, since (0.9)² + (0.9)² = 1.62 >1

c) Cosider the set C = {(x,y) | y>x}

This cannot be written as a Cartesian product of two subsets of R. Suppose C = {(x,y) | y>x} = A x B for some A; B

subsets of R. The points ( 0,1) and (1,2) are in C . Since 1 is in A and 1 is in B sothat (1,1) is in A x B This is not true since ( 1,1) is not in C