Exercise · ZW-ORDERED-FIELD-007 · ★★☆ Intermediate
For \(a,b\) in an ordered field, how many of \(a<b\), \(a=b\), \(a>b\) are true?
Choose one option.
Show answer
- D. Exactly one
Explanation
This is trichotomy: exactly one of the three relations holds.
Review the concept: Ordered field
An ordered field is a field \(F\) with a total order \(<\) compatible with addition and with multiplication by positive elements.
\[a0\Rightarrow ac
More practice problems: ordered field
- ZW-ORDERED-FIELD-001 If a<b and c<0 in an ordered field, what holds?
- ZW-ORDERED-FIELD-002 Which of these sets, with the usual operations and order, is not an ordered field?
- ZW-ORDERED-FIELD-003 Which statement is true in every ordered field F?
- ZW-ORDERED-FIELD-004 If a<b in an ordered field, which inequality follows?
- ZW-ORDERED-FIELD-005 Multiply the inequality -3<2 by -4. What results?
- ZW-ORDERED-FIELD-006 If 0<a<b, which is true?
- ZW-ORDERED-FIELD-008 If 0<a<b in an ordered field, how do a+b and 2a compare?
- ZW-ORDERED-FIELD-009 Why can the finite field mathbb Z/p (p prime) not be made an ordered field?