A function

f:A\rightarrow B

is called injective function (or one one function) if

(\forall x,y\in A)[f(x)=f(y)\Rightarrow x=y]

if and only if

(\forall x,y\in A)[x\neq y\Rightarrow f(x)\neq f(y)]

if and only if

It is common for people who do not engage deeply with mathematics to think of addition only in terms of the everyday operation they know: for example, 1+2=31 + 2 = 3 or 5+3=85 + 3 = 8. This everyday understanding of addition, often called “standard addition,” is straightforward and intuitive. However, in mathematics, it is possible to define alternative types of addition that differ from this standard operation.

For example, consider the set of all integers. On this set, we can define a new addition operation as follows:

• 1+3=71 + 3 = 7,
• 2+4=102 + 4 = 10,
• −7+3=−1-7 + 3 = -1,
  and so on.