Feature that is applied to a set
Informally, a set endowned with a structure or multiple structures of 'geometric nature' which are the primary focus
Informally, a set endowned with a structure or multiple structures of 'algebraic nature' which are the primary focus
Anything with formally defined properites
Subset of \(\mathbb{R}\) categorizing a continuous range of elements
Interval excluding its endpoints
\( (a,b) = \{x \in \mathbb{R} : a \lt x \lt b \}\)
Interval including its endpoints
\( [a,b] = \{x \in \mathbb{R} : a \leq x \leq b \}\)
Set \(X\) representing the space of the inputs of function \(f\)
Subset of \(X\) representing all elements that is a well-defined input for function \(f\)
\(\text{dom}(f) = \{ x \in X : f(x) \text{ is well defined } \}\)
Set \(Y\) representing the space of the output of function \(f\)
Subset of \(Y\) representing all elements mapped to a domain element of function \(f\)
\( \text{im}(f) = f(\text{dom}(f)) \)
Set of all elements in a function's image mapped to elements in \(U\)
\(f(U) = \{ f(u) : u \in U \}\)
Given some function \(f : X \to Y\), an image is a set resolving some or all (possible) codomain elements to domain elements
Set of all domain elements mapped to elements in \(V\)
\( f^{-1}(V) = \{ x : f(x) \in V \}\)
Set of all domain elements mapped to elements in the codomain
\(f^{-1}(f(\text{dom}(f)) = \text{dom}(f) \)
Considering a function \(f : X \to Y\), the inverse image or pre image of some range subset \(V\) is the domain subset the function is mapped to
\( f^{-1}(V) = \{ x : f(x) \in V \}\)