aLet abc and d be real numbers such that a

a)Let a,b,c and d be real numbers such that a<b and c<d. Prove that |[a,b]×[c,d]| = c.

b) Suppose that (a, b) and (c, d) are two distinct points in the Cartesian plane R2 = R × R. Let S be the set of points on the line segment that joins the points (a, b) and (c, d). Prove that |S| = c.

c) Let S be the set of all pairs (x,y) in the Cartesian plane R^2 = R×R satisfying x^2 +y^2 < 1 and x+y > 1. Sketch the set S and prove that |S| = c.

Solution

a) For infinite sets X , the product XxX has the same cardinality as X.

Any closed interval [a,b] is bijective with [0,1] (using the linear map t going to a(1-t) +bt, with t in [0,1]).

So any two closed intervals have the same cardinality, and using tan^-x x on R , we can show that any closed interval has the same cardinality as R, (=c, as denoted here)

(b) Same arguement as in (a). We may consider a =c=0 and b =d=1 and use the mapping given above. The open interval (0,1) has the same cardinality as [0,1] , hence the result.

(c) The required set is the colored region .As S is a subset of [0,1]x[0,1] its cardinality cannot be greater than c. It is cleary uncountable (consider the map [0,1] to the line segment x+y=1 colored green.

Thus Cardinality of S is c