Indirect Arg: Contradiction & Contraposition

Question

Consider the following statement.

There is no greatest negative real number.

Identify which of the following is a negation for the statement and then fill in the blanks to prove the statement by contradiction.

There is a least negative real number.There is no greatest positive real number.     There is a greatest negative real number.There is no greatest negative real number.There is a greatest positive real number.

Proof by contradiction:

Suppose not. That is, suppose  ---Select--- there is a least negative real number a there is no greatest positive real number a there is a greatest negative real number a there is no greatest negative real number a there is a greatest positive real number a . Then, by assumption, a <  

 

 

 and, for every negative real number x, a ≥  

 

 .

Find a new real number, expressed in terms of a, that is both negative and greater than a. Use this expression to fill in the blank in the following inequality.

a <  

 

 < 0

This result contradicts the assumption that a is  ---Select--- the least negative real number not the greatest positive real number the greatest negative real number not the greatest negative real number the greatest positive real number . Therefore the supposition is false and the given statement is true.

Answer 1

Suppose there is a greatest negative real number.

Then there is x < a < 0 such that any real number x < a.

then a/2 < 0

By assumption, a/2 < a < 0 <-- any negative real # must be less than a

a < 2a < 0 <-- multiplies everything by 2

-a < 0 <-- subtracts 2a from both sides

a>0 <-- multiplies by -1, signs change

this is a contradiction since a is negative