Indicate which of the following statements about sets under the specified operations are true. For the ones that are false, provide one counter-example.
(a) The set of irrational numbers under addition forms an abelian group.
(b) The set of complex numbers under multiplication forms a group.
(c) The set of rational numbers under multiplication and addition forms a ring.
(d) The set of rational numbers under multiplication and addition does not form a field.
(a) This statement is false, as the addition of two irrational numbers does not guarantee another irrational number. Ex: (1 + √2) + (1 - √2) = 2, which is not an irrational number.
(b) The ...
This post examines irrational numbers and other concepts.