What is a surjective function? I am a 9th grader self-studying about set theory and functions. I understood most basic concepts, but I didn't understand what is a surjective function. I have understood what is an injective function, and if I know what is a surjective function, I think I could understand what is a bijective function (this is my main goal).
In formal terms a function $f$ from $A$ to $B$ is said to be surjective if for all $y$ in $B$, there exists $x$ in $A$ such that $f(x)=y$. I don't understand this clearly because i'm still new to these notations. Can you explain this in intuitive way? And for example can you give me a surjective function that is not injective, and inversely, and neither one of the two?

Basically, if a function $f: A \to B$ is surjective, it means that every element of the set $B$ can be "obtained" through the function $f$. Formally, like you said, it means that for every $b \in B$, there exists some $a \in A$ such that $f(a) = b$. Phrased differently, $f$ being surjective means that the range of $f$ includes all of the set $B$. 
For an example, take  $A = \{1,2,3,4\}$ and $B = \{x,y,z\}$. Define $f : A \to B$ by $$f(1) = x$$ $$f(2) = y$$ $$f(3) = z$$ $$f(4) = z$$
Then $f$ is surjective because each element of $B$ has a corresponding element of $A$ which can be mapped onto it; but $f$ is not injective since both 3 and 4 are mapped into the same element of $B$, namely $z$.
If you just fool around with simple functions and sets like these, you can easily get some nice examples of the other two cases you wanted.
