Properties of Integer

Closure (闭包)

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• A property of a set whereby an operation on members of the set always produces a member of the same set is called closure. For example, integers are closed under addition and multiplication, like $1+1=2$, where $2$ is an integer (By closure of integers under multiplication and addition)

Commutativity (交换律)

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• The order in which the operands are taken does not affect the result of the operation. Addition and multiplication are commutative, meaning $x+y=y+x$

Associativity (结合律)

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• You do not need to worry about the placement of parentheses in expressions with more than two operands when the operations are associative. For example, addition and multiplication are associative

Distributivity (分配律)

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$$x\times (y+z)=(x\times y)+(x\times z)$$

• Multiplication distributes over addition

Trichotomy (三分法)

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• For any two given inputs ($x$ and $y$), exactly one of the following three outcomes must be true: $x=y$, $x<y$ or $x>y$