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Find a set S of two elements that satisfies the following:

(1) S satisfies all the properties of the field except distributivity.

(2) \exists x \in S, 0x \neq 0.


Let S = \{ a, b \} where a is the additive identity and b is the multiplicative identity and a \neq b. After trial and error, we have the following addition and multiplication tables:

+ a b
a a b
b b a
\times b a
b b a
a a b

We verify that S satisfies (1). By the addition and multiplication tables, then S satisfies closure, commutativity, associativity and existence of identities and inverses. Since a(b + b) = a(a) = b \neq a = a + a = ab + ab, then S does not satisfy distributivity. Then S satisfies (1).

We verify that S satisfies (2). Since aa = b \neq a, then S satisfies (2).

Elementary Errors in Homework

(1) Prove A \implies B. Assume A and derive B. It is not the other way around.

(2) Prove A \iff B. Show that A \implies B and B \implies A.

(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.