Let A be a set with 2 elements, which we call e and o. Let ⊕ and ⊗ be two binary
operations on A. We call ⊕ addition and ⊗ multiplication. These operations are defined by
⊕ e o
e e o
o o e
⊗ e o
e e e
o e o
(For example, to find e ⊗ e, we go to the table on the right above and we see that the entry
in the e column and in the e row is e. So e ⊗ e = e. Think of e as even and o as odd, and
you will see the pattern.)
One can check that A with ⊕ and ⊗ satisfies the field axioms A1 to A4, M1 to M4, and
DL on p.13 in our text. (You do not need to check this.) Identify the multiplicative identity,
the additive identity, and the additive inverse of the multiplicative ident

Given the binary operations \( \oplus \) (addition) and \( \otimes \) (multiplication) defined on the set \( A = \{e, o\} \), we need to identify the multiplicative identity, the additive identity, and the additive inverse for the multiplicative identity.

### 1. **Additive Identity**
The additive identity is an element \( a \in A \) such that for every \( x \in A \), the equation \( a \oplus x = x \oplus a = x \) holds.

Looking at the addition table \( \oplus \):

\[
\begin{array}{c|cc}
\oplus & e & o \\
\hline
e & e & o \\
o & o & e \\
\end{array}
\]

- \( e \oplus e = e \) and \( e \oplus o = o \) 
- \( o \oplus e = o \) and \( o \oplus o = e \)

From the table, \( e \oplus x = x \oplus e = x \) for every \( x \in A \). Hence, **\( e \) is the additive identity**.

### 2. **Multiplicative Identity**
The multiplicative identity is an element \( m \in A \) such that for every \( x \in A \), the equation \( m \otimes x = x \otimes m = x \) holds.

Looking at the multiplication table \( \otimes \):

\[
\begin{array}{c|cc}
\otimes & e & o \\
\hline
e & e & e \\
o & e & o \\
\end{array}
\]

- \( e \otimes e = e \) and \( e \otimes o = e \)
- \( o \otimes e = e \) and \( o \otimes o = o \)

From the table, \( o \otimes o = o \), but \( e \otimes o = e \). However, \( o \otimes e = e \). Therefore, **\( o \) is the multiplicative identity**.

### 3. **Additive Inverse of the Multiplicative Identity**
The additive inverse of an element \( x \) is an element \( y \) such that \( x \oplus y = y \oplus x = e \) (where \( e \) is the additive identity).

The multiplicative identity was identified as \( o \). We need to find \( y \in A \) such that \( o \oplus y = e \).

From the addition table:
- \( o \oplus o = e \)

Thus, the additive inverse of \( o \) is \( o \) itself. Hence, **\( o \) is the additive inverse of the multiplicative identity \( o \)**.

### Summary
- **Additive identity:** \( e \)
- **Multiplicative identity:** \( o \)
- **Additive inverse of the multiplicative identity:** \( o \)

Would you like further details or have any questions about this? Here are 5 related questions to explore further:

1. What are the field axioms A1 to A4, M1 to M4, and DL?
2. How does the concept of an identity element generalize to other algebraic structures?
3. What is the significance of self-inverse elements in algebraic structures?
4. Can you describe a situation where the same element serves as both an additive and multiplicative identity?
5. How can binary operations on finite sets be used to model digital logic or computer science applications?

**Tip:** When analyzing binary operations, always look for identity elements first, as they often provide a foundation for understanding the structure.

Explore how to identify additive and multiplicative identities and inverses in a set with two elements, using binary operations. Learn about the properties of addition and multiplication in this mathematical analysis.

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