Coordinate function

1. Sablon:Matematika 1. $f:R\to \prod _{i\in I}{S}_i$ is a function where the codomain is the product of sets $S_i$ indexed by $I$. So for each $x\in R$, $f(x)$ is an element of the product $\prod _{i\in I}{S}_i$, which is a tuple where the $i$-th component is in $S_i$.

2. For each $j\in I$, $p_j$ is the projection onto the $j$-th component, which means $p_j(f(x))$ selects the $j$-th coordinate of the tuple $f(x)$.

3. The composition $p_j\circ f:R\to S_j$ is a function that maps each $x\in R$ to the $j$-th component of $f(x)$, denoted as $f_j(x)$. Thus, $p_j(f(x))=f_j(x)$, where $f_j$ is the function that returns the $j$-th coordinate of $f(x)$.

In summary, $p_j\circ f$ gives the $j$-th coordinate function of $f$, and it extracts the $j$-th component of the tuple $f(x)$ from the product $\prod _{i\in I}{S}_i$. Sablon:Engl