I'm planning to tackle all my work by learning all the definitions and statements relevant to my subjects by rewriting them in my own style. Below, I've stating the definition of a function in my own way using quantifiers. Can you please tell me if writing in this way is ok and correct? and if it makes logical sense? My work on self-teaching maths has never had the chance to be criticised which doesn't make me as confident as I would like. Also, is learning topics in maths by working with the definitions and important statements first by applying examples, non-examples, counter-examples first, then reading all relevant information in-between and the proofs later a better method of study? 

**Definition**
Let $X$ and $Y$ be sets. A *function* or *mapping* $f$ from a set $X$ to a set $Y$, denoted $f:X\rightarrow Y$, is a rule such that $\forall{x}\in{X}, \exists{y}\in{Y}$ s.t $y=f(x)$.