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 ($\in \mathbb{Z}$).
• 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 ($\in \mathbb{Z}$).
• 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 ($\notin \mathbb{Z}$), and thus breaks the closure property.

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

• Closed: For any real number $a,b\in ℝ$, $a\cdot b=c$ is also a real number ($\in ℝ$).
• Associative: For any real number $a,b,c\in ℝ$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
• Identity: The multiplicative identity is 1, as for any real number $a\in ℝ$, $a\cdot 1=a$.
• Inverse: For each real number $a\in ℝ$ (except for 0, the identity), its multiplicative inverse is ${\displaystyle \frac{1}{a}}$, which is a real number ($\in ℝ$).