Sahithyan's S1 — Mathematics
Field Axioms
Field Axioms of R
with two binary operations and satisfying the following properties
- Closed under addition:
- Commutative:
- Associative:
- Additive identity:
- Additive inverse:
- Closed under multiplication:
- Commutative:
- Associative:
- Multiplicative identity:
- Multiplicative inverse:
- Multiplication is distributive over addition:
Required proofs
The below mentioned propositions can and should be proven using the above-mentioned axioms. .
- Hint: Start with
- Additive identity () is unique
- Multiplicative identity () is unique
- Additive inverse () is unique for a given
- Multiplicative inverse () is unique for a given
Field
Any set satisfying the above axioms with two binary operations (commonly and ) is called a field. Written as:
Field or Not?
Is field? | Reason (if not) | |
---|---|---|
True | ||
False | Axiom 11 is invalid | |
False | Multiplicative inverse doesn’t exist | |
True | ||
False | ||
Boolean algebra | False | Additive inverse doesn’t exist |
True | ||
True | ||
False | Multiplicative inverse doesn’t exist |