Find a set of two elements that satisfies the following:
(1) satisfies all the properties of the field except distributivity.
Let where is the additive identity and is the multiplicative identity and . After trial and error, we have the following addition and multiplication tables:
We verify that satisfies (1). By the addition and multiplication tables, then satisfies closure, commutativity, associativity and existence of identities and inverses. Since , then does not satisfy distributivity. Then satisfies (1).
We verify that satisfies (2). Since , then satisfies (2).
Elementary Errors in Homework
(1) Prove . Assume and derive . It is not the other way around.
(2) Prove . Show that and .
(3) This is for Boris's section only. When a proof requires a previous result, there are two possibilities:
- (a) The result is already proved in class or in a previous homework. Then state the result and use it without proof.
- (b) The result is neither proved in class nor in a previous homework. Then reference it in the textbook or prove it yourself.