Def An
injection is an
injective function f:X-->Y.
That is, given any
x and
y in
X, if we have
f(x)=f(y) then this implies that
x=y. Equivalently, whenever
x and
y in
X are not equal then
f(x) and
f(y)
are not equal.
If you're not familiar with functions, a function f:X-->Y
is a black box
that given as input an element from the set X
gives as output an element f(x) from the set Y.
So to say the function is injective just means that if you have
two distinct inputs then the outputs will also be distinct.
An example of an injection is the identity function 1X:X-->X which takes every element to itself.
Another example (this time not a bijection) is the function
f:Z-->Z with f(n)=2n.
Here Z denotes the set of all integers.
Here's an example that is not an injection. Consider the function
f:Z-->Z with f(n)=n2. This time f is not injective
because f(-1)=f(1).