Properties of Integers
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Additive Group of Integers is Countably Infinite Abelian Group
The set of integers under addition $\struct {\Z, +}$ forms a countably infinite abelian group.
Integer Addition is Closed
The set of integers is closed under addition:
- $\forall a, b \in \Z: a + b \in \Z$
Integer Addition is Associative
The operation of addition on the set of integers $\Z$ is associative:
- $\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$
Integer Addition Identity is Zero
The identity of integer addition is $0$:
- $\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$
Inverse for Integer Addition
Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition:
- $\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$
Integer Addition is Commutative
The operation of addition on the set of integers $\Z$ is commutative:
- $\forall x, y \in \Z: x + y = y + x$
Integers are Countably Infinite
The set $\Z$ of integers is countably infinite.
Integers under Addition form Totally Ordered Group
Then the ordered structure $\struct {\Z, +, \le}$ is a totally ordered group.
Integers under Multiplication form Semigroup
The set of integers under multiplication $\struct {\Z, \times}$ is a semigroup.
Integer Multiplication is Closed
The set of integers is closed under multiplication:
- $\forall a, b \in \Z: a \times b \in \Z$
Integer Multiplication is Associative
The operation of multiplication on the set of integers $\Z$ is associative:
- $\forall x, y, z \in \Z: x \times \paren {y \times z} = \paren {x \times y} \times z$
Integers under Multiplication form Monoid
The set of integers under multiplication $\struct {\Z, \times}$ is a monoid.
Integer Multiplication Identity is One
The identity of integer multiplication is $1$:
- $\exists 1 \in \Z: \forall a \in \Z: a \times 1 = a = 1 \times a$
Integers under Multiplication form Countably Infinite Commutative Monoid
The set of integers under multiplication $\struct {\Z, \times}$ is a countably infinite commutative monoid.
Integer Multiplication is Commutative
The operation of multiplication on the set of integers $\Z$ is commutative:
- $\forall x, y \in \Z: x \times y = y \times x$
Integers form Commutative Ring
The set of integers $\Z$ forms a commutative ring under addition and multiplication.
Integers under Addition form Abelian Group
The set of integers under addition $\struct {\Z, +}$ forms an abelian group.
Integer Multiplication Distributes over Addition
The operation of multiplication on the set of integers $\Z$ is distributive over addition:
- $\forall x, y, z \in \Z: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$
- $\forall x, y, z \in \Z: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$
Integer Multiplication has Zero
The set of integers under multiplication $\struct {\Z, \times}$ has a zero element, which is $0$.
Integers form Commutative Ring with Unity
The integers $\struct {\Z, +, \times}$ form a commutative ring with unity under addition and multiplication.
Integer Multiplication has Identity Element
The identity of integer multiplication is $1$:
- $\exists 1 \in \Z: \forall a \in \Z: a \times 1 = a = 1 \times a$
Integers under Addition form Infinite Cyclic Group
The additive group of integers $\struct {\Z, +}$ is an infinite cyclic group which is generated by the element $1 \in \Z$.
Integers form Integral Domain
The integers $\Z$ form an integral domain under addition and multiplication.
Integers form Commutative Ring with Unity
The integers $\struct {\Z, +, \times}$ form a commutative ring with unity under addition and multiplication.
Ring of Integers has no Zero Divisors
The integers have no zero divisors:
- $\forall x, y, \in \Z: x \times y = 0 \implies x = 0 \lor y = 0$
Integers form Totally Ordered Ring
The structure $\struct {\Z, +, \times, \le}$ is a totally ordered ring.
Integers form Ordered Integral Domain
The integers $\Z$ form an ordered integral domain under addition and multiplication.
Integers are Euclidean Domain
The integers $\Z$ with the mapping $\nu: \Z \to \Z$ defined as:
- $\forall x \in \Z: \map \nu x = \size x$
form a Euclidean domain.
Substructures and Superstructures
Additive Group of Integers is Normal Subgroup of Rationals
Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\Q, +}$ be the additive group of rational numbers.
Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$.
Additive Group of Integers is Normal Subgroup of Reals
Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.
Additive Group of Integers is Normal Subgroup of Complex
Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\C, +}$ be the additive group of complex numbers.
Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\C, +}$.
Integers form Subdomain of Rationals
The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of rational numbers.
Integers form Subdomain of Reals
The integral domain of integers $\struct {\Z, +, \times}$ forms a subdomain of the field of real numbers.