Injective or one-to-one? What is the difference?

Question

What is the difference between the terms 'injective' and 'one-to-one', 'surjective' and 'onto', and 'bijective' and 'isomorphic'?

Answer 1

Injective and one-to-one mean the same thing.

Surjective and onto mean the same thing.

Bijective means both injective and surjective. This means that there is an inverse, in the widest sense of the word (there is a function that "takes you back"). The inverse is so-called two-sided, which means that not only can you go there and back again, but you could also start at the other end, go back and then there again. Either way you end up where you started.

Isomorphism is a bit different. It means that not only is there an inverse, but the inverse is "nice" (and the function itself must be "nice" too). For instance, in topology, we're mostly interested in continuous maps, so "nice" in that setting means "continuous". There are bijective, continuous maps where the inverse is not continuous. Other niceties include "linear" for vector spaces, "differentiable" for calculus / analysis and "homomorphism" in abstract algebra.

Example: you have the function that takes the two intervals $(0, 1)$ and $[2, 3]$ and "glues" them together at one end to make $(0, 2]$. This is a bijection (every element in $(0, 1)$ stays put, and every element in $[2, 3]$ gets sent to $[1, 2]$ by subtracting $1$), so there is an inverse. However, this inverse is not continuous, and therefore the bijection is not an isomorphism. (In topology, isomorphisms are usually called homeomorphisms instead.)

Another example: You can find a bijection from $\Bbb R$ to $\Bbb R^2$ (google "plane filling curves" to see some surjections, that you can find an injection is obvious enough, and with some mathematical mumbo-jumbo, you can combine the two to make a bijection). But you'd have to struggle to make it continuous, and it is provably impossible to make both the function and the inverse continuous.

Note: the moment you start mixing in the word "morphism" in anything, there is always this "nicety" condition implied (a "morphism" is a nice function, whatever that means in your setting). This is from the part of mathematics called category theory, where you try to study functions without ever mentioning the elements of your sets. You often hear things like "there is a morphism here such that for every morphism there, there is a unique morphism..." and so on.

Answer 2

Let $f:A\to B$

Injective/one-to-one: $f(x)=f(y)\implies x=y$

Surjective/onto: for all $b\in B$, there is an $x\in A$ with $f(x)=b$ (basically, all elements of the codomain are represented by $f$)

Bijective: $f$ is both injective and surjective

Isomorphism: a bijective map between two algebraic structures

Answer 3

Injective and Surjective has been given already, the differens between isomorphism and bijection is that for a bijection $f$ versus an isomorphism $\phi$ we have that $\phi(a\cdot b)=\phi(a)\cdot\phi(b)$ while for for $f$ do we not have it, for a bijection we can have $f(a\cdot b)\neq f(a)\cdot f(b)$

Examples of them are $f:R_+\to R_+$ where $f(x)=x^2$, this is a bijection but $$f(a+b)=(a+b)^2 =a^2+2ab+b^2 \neq a^2+b^2 = f(a) + f(b)$$

Let $\phi:Z\to 2Z$ be an isomorphism with $\phi(a)=2a$ then $$\phi(a+b)=2(a+b)=2a+2b=\phi(a)+\phi(b)$$