Mark each statement True of False a Each nonempty subset of

Mark each statement True of False. (a) Each nonempty subset of R contains a minimum. (b) Each bounded subset of R has a least upper bound. (c) If E is a closed subset in R then R\\E is open. (d) Between any two distinct real numbers the there is a rational number.

Solution

(a). The statement is False. The non-empty subset (-,0] of R has no minimum.

(b) The statement is True. By definition, a set X R of real numbers is bounded from above if there exists a real number r R, called an upper bound of X, such that x r for every x X.

(c ) The statement is True. R = (-,) is an open interval of real numbers. If [x,y] R, then R\\E = (-,x)U(y, ) is open.

(d) The statement is True. If a and b are 2 distinct real numbers and a < b, then (a+b) is a rational number between a and b.