I have read that in the ordered pair $z=(x_1,x_2)$, an element of a direct product $Z=X_1 \times X_2$ of sets $X_1$ and $X_2$, the element $x_1$ is called the first projection and $x_2$ is called the second projection, is this just by definition, or is there a more meaningful reason for these names?
they can then be denoted pr$_1z$ and pr$_2z$
should i just accept this, or maybe i can think of it in a different way to being just names?
p.s. i looked up the mathjax stuff for this, tell me if it is alright, thanks!
It is from the definition. However you may be intereseting in that what it come from or why we called it by this name.
In fact: Pick a point $(x,y)$ from the space $\Bbb R^2$, you will find that $x$ is just the projection of the point $(x,y)$, and $y$ is also!