# 14-240/Tutorial-Sep30

## Boris

#### Problem

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

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

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

Solution:

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.