Math 120A HW 2

Due Wednesday, January 21.
  1. 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.
  2. 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.)
  3. For each of the following functions, say whether it is a homomorphism and also whether it is an isomorphism. Justify your answers.
    1. 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 $(\mathbb{Z}, +)$?
    2. 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 $(\mathbb{R}, \cdot)$?
    3. 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 $(\mathbb{R}, +)$?
  4. 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.
  5. 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.
  6. Using the structural property from the previous problem, prove the following statements:
    1. $(\mathbb{Z},+) \not\cong (\mathbb{Q}, +)$.
    2. $(\mathbb{Q},+) \not\cong (\mathbb{Q},\cdot)$.