Decide whether the following statements are true or false. If the statement is false give a counterexample, otherwise leave the field empty.
(1) Let the binary operation $\star: \mathbb{Z}_{3} \times \mathbb{Z}_{3} \rightarrow \mathbb{Z}_{3}$ be given by $a \star b=\left(a^{b}\right) \bmod 3$.
The statement is false. $\checkmark$ The identity element with respect to $\star$ is 0 .
Counterexample: The statement is false, because for $b:-1 \quad \in \mathbb{Z}_{3}$ we have $0 * b \neq b$.
(2) Let the binary operation $\Theta: \mathbb{Z}_{13} \times \mathbf{Z}_{13} \rightarrow \mathbb{Z}_{13}$ be given by $a \ominus b=(a-b)$ mod 13
The statement is faise. $\checkmark$ The identity element with respect to $\theta$ is 1
Counterexample: The statement is false, because for $b:=13 \quad \in \mathbf{Z}_{13}$ we have $1 \ominus b \neq b$.
(3) Let the binary operation $\star: \mathbb{Z}_{10} \times \mathbb{Z}_{10} \rightarrow \mathbb{Z}_{10}$ be given by $a * b=\left(a^{b}\right) \bmod 10$.
The statement is false: $\checkmark$ The identity element with respect to $I_{L} 4$
Counterexample: The statement is false, because for $b:=1 \quad \in \mathbf{Z}_{40}$ we have $4 * b \neq b$.
Answer
Final Answer: The statements are false. The counterexamples are as follows: \[\boxed{1. \text{For the operation } a \star b=\left(a^{b}\right) \bmod 3, \text{ the identity element is not 0. A counterexample is } 0 \star 2 = 0 \neq 2.} \] \[\boxed{2. \text{For the operation } a \ominus b=(a-b) \text{ mod 13, the identity element is not 1. A counterexample is } 1 \ominus 13 = 1 \neq 13.} \] \[\boxed{3. \text{For the operation } a * b=\left(a^{b}\right) \bmod 10, \text{ the identity element is not 4. A counterexample is } 4 * 1 = 4 \neq 1.} \]
Step 1 ：Decide whether the following statements are true or false. If the statement is false give a counterexample, otherwise leave the field empty.
Step 2 ：(1) Let the binary operation $\star: \mathbb{Z}_{3} \times \mathbb{Z}_{3} \rightarrow \mathbb{Z}_{3}$ be given by $a \star b=\left(a^{b}\right) \bmod 3$. The statement is false. The identity element with respect to $\star$ is 0. Counterexample: The statement is false, because for $b:-1 \quad \in \mathbb{Z}_{3}$ we have $0 * b \neq b$.
Step 3 ：(2) Let the binary operation $\Theta: \mathbb{Z}_{13} \times \mathbf{Z}_{13} \rightarrow \mathbb{Z}_{13}$ be given by $a \ominus b=(a-b)$ mod 13. The statement is false. The identity element with respect to $\theta$ is 1. Counterexample: The statement is false, because for $b:=13 \quad \in \mathbf{Z}_{13}$ we have $1 \ominus b \neq b$.
Step 4 ：(3) Let the binary operation $\star: \mathbb{Z}_{10} \times \mathbb{Z}_{10} \rightarrow \mathbb{Z}_{10}$ be given by $a * b=\left(a^{b}\right) \bmod 10$. The statement is false. The identity element with respect to $I_{L} 4$. Counterexample: The statement is false, because for $b:=1 \quad \in \mathbf{Z}_{40}$ we have $4 * b \neq b$.
Step 5 ：Final Answer: The statements are false. The counterexamples are as follows: \[\boxed{1. \text{For the operation } a \star b=\left(a^{b}\right) \bmod 3, \text{ the identity element is not 0. A counterexample is } 0 \star 2 = 0 \neq 2.} \] \[\boxed{2. \text{For the operation } a \ominus b=(a-b) \text{ mod 13, the identity element is not 1. A counterexample is } 1 \ominus 13 = 1 \neq 13.} \] \[\boxed{3. \text{For the operation } a * b=\left(a^{b}\right) \bmod 10, \text{ the identity element is not 4. A counterexample is } 4 * 1 = 4 \neq 1.} \]
