A-2.2 Examples

$\mathbb{Z}$ (i.e., the set of all integers) is an abelian group under addition ($+$), because:

• Closed: For any integer $a,b\in \mathbb{Z}$, $a+b=c$ is also an integer (i.e. $a+b\in \mathbb{Z}$).
• Associative: For any integer $a,b,c\in \mathbb{Z}$, $(a+b)+c=a+(b+c)$.
• Identity: The additive identity is 0 because, for any $a\in \mathbb{Z}$, $a+0=a$.
• Inverse: For each $a\in \mathbb{Z}$, its additive inverse is $-a$, as $a+(-a)=0$.
• Commutative: For any integer $a,b\in \mathbb{Z}$, $a+b=b+a$.

$\mathbb{Z}$ is a monoid under multiplication ($\cdot$) because:

• Closed: For any integer $a,b\in \mathbb{Z}$, $a\cdot b=c$ is also an integer (i.e., $a\cdot b\in \mathbb{Z}$).
• Associative: For any integer $a,b,c\in \mathbb{Z}$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
• Identity: The multiplicative identity is 1, because for any $a\in \mathbb{Z}$, $a\cdot 1=a$.
• NO Inverse: For an integer $a\in \mathbb{Z}$, its multiplicative inverse is ${\displaystyle \frac{1}{a}}$, but this is not necessarily an integer ($\notin \mathbb{Z}$); therefore, not every element has a multiplicative inverse. Thus, $(\mathbb{Z},\cdot )$ is not a group (though it is a monoid).

${ℝ}^{\times }$ (i.e., the set of all nonzero real numbers) is an abelian group under multiplication ($\cdot$), because:

• Closed: For any real number $a,b\in {ℝ}^{\times }$, $a\cdot b=c$ is also a real number (and remains in ${ℝ}^{\times }$).
• Associative: For any real number $a,b,c\in {ℝ}^{\times }$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
• Identity: The multiplicative identity is 1, as for any real number $a\in {ℝ}^{\times }$, $a\cdot 1=a$.
• Inverse: For each real number $a\in {ℝ}^{\times }$, its multiplicative inverse is ${\displaystyle \frac{1}{a}}$, which is a non-zero real number ($\in {ℝ}^{\times }$).