Math 120A HW 2
Due Wednesday, January 21.
- Let $S = \{a,b,c\}$ where $a$, $b$, and $c$ are distinct elements. Consider the binary operation $\ast$ defined on $S$ by the following table:
\begin{array}{c|ccc}
\ast & a & b & c\\
\hline
a & a & a & a\\
b & a & b & c\\
c & c & c & c
\end{array}
Is the operation commutative? Does the binary structure $(S,\ast)$ have an identity element? Justify your answers.
- Let $S = \{0,1\}$ and consider the binary operations $\ast$ and $\ast'$ defined on $S$ by the following tables:
$$
\begin{array}{c|cc}
\ast & 0 & 1\\
\hline
0 & 0 & 0\\
1 & 0 & 1
\end{array}
\quad
\begin{array}{c|cc}
\ast' & 0 & 1\\
\hline
0 & 0 & 1\\
1 & 1 & 1
\end{array}
$$
Are the binary structures $(S,\ast)$ and $(S,\ast')$ isomorphic? Justify
your answer. (Note: saying that two structures are isomorphic means
the same thing as saying that there is an isomorphism from one to the other.
Recall that this is a symmetric notion.)
- For each of the following functions, say whether it is a homomorphism and also whether it is an isomorphism. Justify your answers.
- Is the function $\phi : \mathbb{Z} \to \mathbb{Z}$ defined by $\phi(n) = 3n$ a homomorphism from $(\mathbb{Z}, +)$ to $(\mathbb{Z}, +)$? Is it an isomorphism from $(\mathbb{Z}, +)$ to $(\mathbb{Z}, +)$?
- Is the function $\phi : \mathbb{R} \to \mathbb{R}$ defined by $\phi(x) = x^3$ a homomorphism
from $(\mathbb{R}, \cdot)$ to $(\mathbb{R}, \cdot)$? Is it an isomorphism from $(\mathbb{R}, \cdot)$ to $(\mathbb{R}, \cdot)$?
- Is the function $\phi : \mathbb{R} \to \mathbb{R}$ defined by $\phi(x) = x^3$ a homomorphism
from $(\mathbb{R}, +)$ to $(\mathbb{R}, +)$? Is it an isomorphism from $(\mathbb{R}, +)$ to $(\mathbb{R}, +)$?
- Prove that the structure $(\mathbb{R}^*,\cdot)$ is isomorphic to the structure $(S,\ast)$ where $S$ denotes the set of all $2 \times 2$ diagonal matrices with real entries and determinant 1, and $\ast$ denotes matrix multiplication. As usual, $\mathbb{R}^*$ denotes the set of all nonzero real numbers.
- Letting $(S,\ast)$ denote an arbitrary binary structure, prove that the property "for every element $b\in S$ there is an element $a \in S$ such that $a \ast a = b$" is a structural property.
- Using the structural property from the previous problem, prove the following statements:
- $(\mathbb{Z},+) \not\cong (\mathbb{Q}, +)$.
- $(\mathbb{Q},+) \not\cong (\mathbb{Q},\cdot)$.