A function f:A –> B is called injective function (or one one function) if for any x and y in A it follows that f(x)=f(y) implies x=y.

Other than the above definition, there are several equivalent definitions. Here they are.

The first one is of course the contraposition of the previous one.

A function f:A–> B is called injective function (or one one function) if for any x and y in A it follows that x≠y implies f(x)≠f(y).

The second one is:

A function f:A–> B is called injective function (or one one function) if for any X,Y⊆A with latex X⊆Y it follows that f(Y\X)=f(Y)\ f(X).

The third one is:

A function f:A–> B is called injective function (or one one function) if for any X,Y ⊆ A, if X∩ Y=Ø, then f(x)∩f(Y)=Ø.

The fourth one is:

A function f:A–> B is called injective function (or one one function) if for any X, Y ⊆ A, it follows that f(X ∩ Y)=f(X) ∩ f(Y).

The fifth one is:

A function f:A–> B is called injective function (or one one function) if for any X⊆A, it follows that f^{-1}(f(X))=X with f^{-1}(Y)={x∈A|f(x)∈ Y} for any Y ⊆ B.

The sixth one is:

A function f:A–> B is called injective function (or one one function) if there exists a function g:B–>A, such that g• f=id_A, with id_A is the identity function on A.

The proof that they are equivalent are left as exercise!!!

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.

From these examples, one might notice a pattern and propose a general rule: for any integers xx and yy, we define x+yx + y as x+y+yx + y + y. Here, the operation still produces an integer, which ensures consistency with the original set of integers.

However, this notation can be confusing because the “+” symbol is being used in two different contexts. On the left side, “+” represents the newly defined operation, while on the right side, “+” refers to standard addition. To avoid this ambiguity, we can replace the “+” symbol for the new operation with another symbol, such as ““. Thus, we can rewrite the rule as x∗y=x+y+yx * y = x + y + y, where “” denotes the newly defined operation.

Using this new operation, we would compute:

• 1∗2=51 * 2 = 5,
• 3∗−1=13 * -1 = 1,
  and so on.

This new operation is just one example of how addition can be redefined. The specific definition can vary as long as the results remain within the domain specified at the outset (in this case, the set of integers). This flexibility allows mathematicians to explore and construct systems with alternative rules, leading to insights and applications beyond standard arithmetic.