Tell whether or not the following sets are fields under the operations + and *. If the set is not a field, tell which one(s) of the Field Axioms do not apply:
a) [rational numbers]
b) [integers]
c) [positive number]
d) non-negative numbers]
That is a question from my text. Can someone please explain this to me? My teacher said something about 44 examples (because there are 11 field axioms).
Eleven seems rather a lot: according to Wolfram MathWorld (I can't post the address, it seems, but just google "field axioms" to find it) there are five: Associativity, Commutativity, Distributivity, Identity and Inverses. Algebraic definitions of these are given there.
The same site supplies this information about fields: "A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra". Can you apply these to your four sets?
Sure! I'll explain each set and whether or not it is a field under the operations + and *.
a) Rational Numbers:
The set of rational numbers is a field under the operations + (addition) and * (multiplication). This set satisfies all 11 field axioms, including closure, associativity, commutativity, existence of additive and multiplicative identities, existence of additive inverses, existence of multiplicative inverses (except for 0), and distributivity.
b) Integers:
The set of integers is not a field under the operations + and *. The main reason is that it does not have multiplicative inverses for every element except for 0. For example, the element 2 does not have a multiplicative inverse in the set of integers.
c) Positive Numbers:
The set of positive numbers is not a field under the operations + and *. The main reason is that it does not have additive inverses for every element. For example, the element 1 does not have an additive inverse in the set of positive numbers.
d) Non-negative Numbers:
The set of non-negative numbers is not a field under the operations + and *. The main reason is that it does not have additive inverses for every element. For example, the element 0 does not have an additive inverse in the set of non-negative numbers.
To determine whether a set is a field or not, you need to check if it satisfies all 11 field axioms. The axioms include properties like closure, associativity, commutativity, existence of identities, existence of inverses, and distributivity. If any of these axioms fail to hold for a particular set, then that set is not a field under the given operations. In this case, you would need to identify which specific axiom(s) are not satisfied.
As for the mentioning of 44 examples, it is unclear why your teacher mentioned that. It might be related to providing examples of sets that are fields or not fields. It could be helpful to ask your teacher for further clarification on that statement.